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3.
Preface
This book is designed for a one semester course in discrete mathematics
for sophomore or junior level students. The text covers the mathematical
concepts that students will encounter in many disciplines such as computer
science, engineering, Business, and the sciences.
Besides reading the book, students are strongly encouraged to do all the
excercises. Mathematics is a discipline in which working the problems is es-
sential to the understanding of the material contained in this book. Students
are encouraged ﬁrst to do the problems without referring to the solutions.
Solutions to problems found at the end of this book can only be used when
you are stuck. Exert a reasonable amount of eﬀorts towards solving a prob-
lem before you look up the answer, and rework any problem you miss.
Students are strongly encouraged to keep up with the exercises and the se-
quel of concepts as they are going along, for mathematics builds on itself.
Solution Manual for the text can be requested from the author through
email:mﬁnan@atu.edu
Marcel B. Finan
May 2001
3
7.
Fundamentals of Mathematical
Logic
Logic is commonly known as the science of reasoning. The emphasis here
will be on logic as a working tool. We will develop some of the symbolic
techniques required for computer logic. Some of the reasons to study logic
are the following:
• At the hardware level the design of ’logic’ circuits to implement in-
structions is greatly simpliﬁed by the use of symbolic logic.
• At the software level a knowledge of symbolic logic is helpful in the
design of programs.
7
8.
8 FUNDAMENTALS OF MATHEMATICAL LOGIC
1 Propositions and Related Concepts
A proposition is any meaningful statement that is either true or false, but
not both. We will use lowercase letters, such as p, q, r, · · · , to represent
propositions. We will also use the notation
p:1+1=3
to deﬁne p to be the proposition 1 + 1 = 3. The truth value of a proposition
is true, denoted by T, if it is a true statement and false, denoted by F, if it
is a false statement. Statements that are not propositions include questions
and commands.
Example 1.1
Which of the following are propositions? Give the truth value of the propo-
sitions.
a. 2 + 3 = 7.
b. Julius Ceasar was president of the United States.
c. What time is it?
d. Be quiet !
Solution.
a. A proposition with truth value (F).
b. A proposition wiht truth value (F).
c. Not a proposition since no truth value can be assigned to this statement.
d. Not a proposition
Example 1.2
Which of the following are propositions? Give the truth value of the propo-
sitions.
a. The diﬀerence of two primes.
b. 2 + 2 = 4.
c. Washington D.C. is the capital of New York.
d. How are you?
Solution.
a. Not a proposition.
b. A proposition with truth value (T).
c. A proposition with truth value (F).
9.
1 PROPOSITIONS AND RELATED CONCEPTS 9
d. Not a proposition
New propositions called compound propositions or propositional func-
tions can be obtained from old ones by using symbolic connectives which
we discuss next. The propositions that form a propositional function are
called the propositional variables.
Let p and q be propositions. The conjunction of p and q, denoted p ∧ q, is
the proposition: p and q. This proposition is deﬁned to be true only when
both p and q are true and it is false otherwise. The disjunction of p and
q, denoted p ∨ q, is the proposition: p or q. The ’or’ is used in an inclusive
way. This proposition is false only when both p and q are false, otherwise it
is true.
Example 1.3
Let
p: 5<9
q : 9 < 7.
Construct the propositions p ∧ q and p ∨ q.
Solution.
The conjunction of the propositions p and q is the proposition
p ∧ q : 5 < 9 and 9 < 7.
The disjunction of the propositions p and q is the proposition
p ∨ q : 5 < 9 or 9 < 7
Example 1.4
Consider the following propositions
p: It is Friday
q: It is raining.
Construct the propositions p ∧ q and p ∨ q.
10.
10 FUNDAMENTALS OF MATHEMATICAL LOGIC
Solution.
The conjunction of the propositions p and q is the proposition
p ∧ q : It is F riday and it is raining.
The disjunction of the propositions p and q is the proposition
p ∨ q : It is F riday or It is raining
A truth table displays the relationships between the truth values of propo-
sitions. Next, we display the truth tables of p ∧ q and p ∨ q.
p q p∧q
T T T
T F F
F T F
F F F
p q p∨q
T T T
T F T
F T T
F F F
Let p and q be two propositions. The exclusive or of p and q, denoted p ⊕ q,
is the proposition that is true when exactly one of p and q is true and is false
otherwise. The truth table of the exclusive ’or’ is displayed below
p q p⊕q
T T F
T F T
F T T
F F F
Example 1.5
a. Construct a truth table for (p ⊕ q) ⊕ r.
b. Construct a truth table for p ⊕ p.
Solution.
a.
11.
1 PROPOSITIONS AND RELATED CONCEPTS 11
p q r p⊕q (p ⊕ q) ⊕ r
T T T F T
T T F F F
T F T T F
T F F T T
F T T T F
F T F T T
F F T F T
F F F F F
b.
p p⊕p
T F
F F
The ﬁnal operation on a proposition p that we discuss is the negation of p.
The negation of p, denoted ∼ p, is the proposition not p. The truth table of
∼ p is displayed below
p ∼p
T F
F T
Example 1.6
Consider the following propositions:
p: Today is Thursday.
q: 2 + 1 = 3.
r: There is no pollution in New Jersey.
Construct the truth table of [∼ (p ∧ q)] ∨ r.
Solution.
p q r p ∧q ∼ (p ∧ q) [∼ (p ∧ q)] ∨ r
T T T T F T
T T F T F F
F T T F T T
F T F F T T
12.
12 FUNDAMENTALS OF MATHEMATICAL LOGIC
Example 1.7
Find the negation of the proposition p : −5 < x ≤ 0.
Solution.
The negation of p is the proposition ∼ p : x > 0 or x ≤ −5
A compound proposition is called a tautology if it is always true, regardless
of the truth values of the basic propositions which comprise it.
Example 1.8
a. Construct the truth table of the proposition (p∧q)∨(∼ p∨ ∼ q). Determine
if this proposition is a tautology.
b. Show that p∨ ∼ p is a tautology.
Solution.
a.
p q ∼p ∼q ∼ p∨ ∼ q p∧q (p ∧ q) ∨ (∼ p∧ ∼ q)
T T F F F T T
T F F T T F T
F T T F T F T
F F T T T F T
Thus, the given proposition is a tautology.
b.
p ∼p p∨ ∼ p
T F T
F T T
Again, this proposition is a tautology.
Two propositions are equivalent if they have exactly the same truth values
under all circumstances. We write p ≡ q.
Example 1.9
a. Show that ∼ (p ∨ q) ≡∼ p∧ ∼ q.
b. Show that ∼ (p ∧ q) ≡∼ p∨ ∼ q.
c. Show that ∼ (∼ p) ≡ p.
a. and b. are known as DeMorgan’s laws.
13.
1 PROPOSITIONS AND RELATED CONCEPTS 13
Solution.
a.
p q ∼p ∼q p∨q ∼ (p ∨ q) ∼ p∧ ∼ q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T
b.
p q ∼p ∼q p∧q ∼ (p ∧ q) ∼ p∨ ∼ q
T T F F T F F
T F F T F T T
F T T F F T T
F F T T F T T
c.
p ∼p ∼ (∼ p)
T F T
F T F
Example 1.10
a. Show that p ∧ q ≡ q ∧ p and p ∨ q ≡ q ∨ p.
b. Show that (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) and (p ∧ q) ∧ r ≡ p ∧ (q ∧ r).
c. Show that (p ∧ q) ∨ r ≡ (p ∨ r) ∧ (q ∨ r) and (p ∨ q) ∧ r ≡ (p ∧ r) ∨ (q ∧ r).
Solution.
a.
p q p∧q q∧p
T T T T
T F F F
F T F F
F F F F
p q p∨q q∨p
T T T T
T F T T
F T T T
F F F F
14.
14 FUNDAMENTALS OF MATHEMATICAL LOGIC
b.
p q r p∨q q∨r (p ∨ q) ∨ r p ∨ (q ∨ r)
T T T T T T T
T T F T T T T
T F T T T T T
T F F T F T T
F T T T T T T
F T F T T T T
F F T F T T T
F F F F F F F
p q r p∧q q∧r (p ∧ q) ∧ r p ∧ (q ∧ r)
T T T T T T T
T T F T F F F
T F T F F F F
T F F F F F F
F T T F T F F
F T F F F F F
F F T F F F F
F F F F F F F
c.
p q r p∧q p∨r q∨r (p ∧ q) ∨ r (p ∨ r) ∧ (q ∨ r)
T T T T T T T T
T T F T T T T T
T F T F T T T T
T F F F T F F F
F T T F T T T T
F T F F F T F F
F F T F T T T T
F F F F F F F F
15.
1 PROPOSITIONS AND RELATED CONCEPTS 15
p q r p∨q p∧r q∧r (p ∨ q) ∧ r (p ∧ r) ∨ (q ∧ r)
T T T T T T T T
T T F T F F F F
T F T T T F T T
T F F T F F F F
F T T T F T T T
F T F T F F F F
F F T F F F F F
F F F F F F F F
Example 1.11
Show that ∼ (p ∧ q) ≡∼ p∧ ∼ q
Solution.
We will use truth tables to prove the claim.
p q ∼p ∼q p∧q ∼ (p ∧ q) ∼ p∧ ∼ q
T T F F T F F
T F F T F T = F
F T T F F T = F
F F T T F T T
A compound proposition that has the value F for all possible values of the
propositions in it is called a contradiction.
Example 1.12
Show that the proposition p∧ ∼ p is a contradiction.
Solution.
p ∼p p∧ ∼ p
T F F
F T F
In propositional functions, the order of operations is that ∼ is performed
ﬁrst. The operations ∨ and ∧ are executed in any order.
16.
16 FUNDAMENTALS OF MATHEMATICAL LOGIC
Review Problems
Problem 1.1
Indicate which of the following sentences are propositions.
a. 1,024 is the smallest four-digit number that is perfect square.
b. She is a mathematics major.
c. 128 = 26
d. x = 26 .
Problem 1.2
Consider the propositions:
p: Juan is a math major.
q: Juan is a computer science major.
Use symbolic connectives to represent the proposition ”Juan is a math major
but not a computer science major.”
Problem 1.3
In the following sentence is the word ”or” used in its inclusive or exclusive
sense? ”A team wins the playoﬀs if it wins two games in a row or a total of
three games.”
Problem 1.4
Write the truth table for the proposition: (p ∨ (∼ p ∨ q))∧ ∼ (q∧ ∼ r).
Problem 1.5
Let t be a tautology. Show that p ∨ t ≡ t.
Problem 1.6
Let c be a contradiction. Show that p ∨ c ≡ p.
Problem 1.7
Show that (r ∨ p) ∧ [(∼ r ∨ (p ∧ q)) ∧ (r ∨ q)] ≡ p ∧ q.
Problem 1.8
Use De Morgan’s laws to write the negation for the proposition:”This com-
puter program has a logical error in the ﬁrst ten lines or it is being run with
an incomplete data set.”
17.
1 PROPOSITIONS AND RELATED CONCEPTS 17
Problem 1.9
Use De Morgan’s laws to write the negation for the proposition:”The dollar
is at an all-time high and the stock market is at a record low.”
Problem 1.10
Assume x ∈ IR. Use De Morgan’s laws to write the negation for the proposition:0 ≥
x > −5.
Problem 1.11
Show that the proposition s = (p ∧ q) ∨ (∼ p ∨ (p∧ ∼ q)) is a tautology.
Problem 1.12
Show that the proposition s = (p∧ ∼ q) ∧ (∼ p ∨ q) is a contradiction.
Problem 1.13
a. Find simpler proposition forms that are logically equivalent to p ⊕ p and
p ⊕ (p ⊕ p).
b. Is (p ⊕ q) ⊕ r ≡ p ⊕ (q ⊕ r)? Justify your answer.
c. Is (p ⊕ q) ∧ r ≡ (p ∧ r) ⊕ (q ∧ r)? Justify your answer.
Problem 1.14
Show the following:
a. p ∧ t ≡ p, where t is a tautology.
b. p ∧ c ≡ c, where c is a contradiction.
c. ∼ t ≡ c and ∼ c ≡ t.
d. p ∨ p ≡ p and p ∧ p ≡ p.
18.
18 FUNDAMENTALS OF MATHEMATICAL LOGIC
2 Conditional and Biconditional Propositions
Let p and q be propositions. The implication p → q is the the proposition
that is false only when p is true and q is false; otherwise it is true. p is called
the hypothesis and q is called the conclusion. The connective → is called
the conditional connective.
Example 2.1
Construct the truth table of the implication p → q.
Solution.
The truth table is
p q p→q
T T T
T F F
F T T
F F T
Example 2.2
Show that p → q ≡∼ p ∨ q.
Solution.
p q ∼p p→q ∼p∨q
T T F T T
T F F F F
F T T T T
F F T T T
It follows from the previous example that the proposition p → q is always
true if the hypothesis p is false, regardless of the truth value of q. We say
that p → q is true by default or vacuously true.
In terms of words the proposition p → q also reads:
(a) if p then q.
(b) p implies q.
(c) p is a suﬃcient condition for q.
(d) q is a necessary condition for p.
(e) p only if q.
19.
2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 19
Example 2.3
Use the if-then form to rewrite the statement ”I am on time for work if I
catch the 8:05 bus.”
Solution.
If I catch the 8:05 bus then I am on time for work
In propositional functions that invlove the connectives ∼, ∧, ∨, and → the
order of operations is that ∼ is performed ﬁrst and → is performed last.
Example 2.4
a. Show that ∼ (p → q) ≡ p∧ ∼ q.
b. Find the negation of the statement ” If my car in the repair shop, then I
cannot go to class.”
Solution.
a. We use De Morgan’s laws as follows.
∼ (p → q) ≡ ∼ (∼ p ∨ q)
≡ ∼ (∼ p)∧ ∼ q
≡ p∧ ∼ q.
b. ”My car in the repair shop and I can get to class.”
The converse of p → q is the proposition q → p. The opposite or in-
verse of p → q is the proposition ∼ p →∼ q. The contrapositive of p → q
is the proposition ∼ q →∼ p.
Example 2.5
Find the converse, opposite, and the contrapositive of the implication: ” If
today is Thursday, then I have a test today.”
Solution.
The converse: If I have a test today then today is Thursday.
The opposite: If today is not Thursday then I don’t have a test today.
The contrapositive: If I don’t have a test today then today in not Thursday
Example 2.6
Show that p → q ≡∼ q →∼ p.
20.
20 FUNDAMENTALS OF MATHEMATICAL LOGIC
Solution.
We use De Morgan’s laws as follows.
p→q ≡ ∼p∨q
≡ ∼ (p∧ ∼ q)
≡ ∼ (∼ q ∧ p)
≡ ∼∼ q∨ ∼ p
≡ q∨ ∼ p
≡ ∼ q →∼ p
Example 2.7
Using truth tables show the following:
a. p → q ≡ q → p
b. p → q ≡∼ p →∼ q
Solution.
a. It suﬃces to show that ∼ p ∨ q ≡∼ q ∨ p.
p q ∼p ∼q ∼p∨q ∼q∨p
T T F F T T
T F F T F = T
F T T F T = F
F F T T T T
b. We will show that ∼ p ∨ q ≡ p∨ ∼ q.
p q ∼p ∼q ∼p∨q p∨ ∼ q
T T F F T T
T F F T F = T
F T T F T = F
F F T T T T
Example 2.8
Show that ∼ q →∼ p ≡ p → q
Solution.
We use De Morgan’s laws as follows.
∼ q →∼ p ≡ q∨ ∼ p
≡ ∼ (∼ q ∧ p)
≡ ∼ (p∧ ∼ q)
≡ ∼ p∨ ∼∼ q
≡ ∼p∨q
≡ p→q
21.
2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 21
The biconditional proposition of p and q, denoted by p ↔ q, is the propo-
sitional function that is true when both p and q have the same truth values
and false if p and q have opposite truth values. Also reads, ”p if and only if
q” or ”p is a necessary and suﬃcient condition for q.”
Example 2.9
Construct the truth table for p ↔ q.
Solution.
p q p↔q
T T T
T F F
F T F
F F T
Example 2.10
Show that the biconditional proposition of p and q is logically equivalent to
the conjunction of the conditional propositions p → q and q → p.
Solution.
p q p→q q→p p↔q (p → q) ∧ (q → p)
T T T T T T
T F F T F F
F T T F F F
F F T T T T
The order of operations for the ﬁve logical connectives is as follows:
1. ∼
2. ∧, ∨ in any order.
3. →, ↔ in any order.
22.
22 FUNDAMENTALS OF MATHEMATICAL LOGIC
Review Problems
Problem 2.1
Rewrite the following proposition in if-then form: ” This loop will repeat
exactly N times if it does not contain a stop or a go to.”
Problem 2.2
Construct the truth table for the proposition: ∼ p ∨ q → r.
Problem 2.3
Construct the truth table for the proposition: (p → r) ↔ (q → r).
Problem 2.4
Write negations for each of the following propositions. (Assume that all vari-
ables represent ﬁxed quantities or entities, as appropriate.)
a. If P is a square, then P is a rectangle.
b. If today is Thanksgiving, then tomorrow is Friday.
c. If r is rational, then the decimal expansion of r is repeating.
d. If n is prime, then n is odd or n is 2.
e. If x ≥ 0, then x > 0 or x = 0.
f. If Tom is Ann’s father, then Jim is her uncle and Sue is her aunt.
g. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3.
Problem 2.5
Write the contrapositives for the propositions of Exercise 40.
Problem 2.6
Write the converse and inverse for the propositions of Exercise 40.
Problem 2.7
Use the contrapositive to rewrite the proposition ” The Cubs will win the
penant only if they win tomorrow’s game” in if-then form in two ways.
Problem 2.8
Rewrite the proposition :” Catching the 8:05 bus is suﬃcient condition for
my being on time for work” in if-then form.
23.
2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 23
Problem 2.9
Use the contrapositive to rewrite the proposition ” being divisible by 3 is a
necessary condition for this number to be divisible by 9” in if-then form in
two ways.
Problem 2.10
Rewrite the proposition ”A suﬃcient condition for Hal’s team to win the
championship is that it wins the rest of the games” in if-then form.
Problem 2.11
Rewrite the proposition ”A necessary condition for this computer program
to be correct is that it not produce error messages during translation” in
if-then form.
24.
24 FUNDAMENTALS OF MATHEMATICAL LOGIC
3 Rules of Inferential Logic
The main concern of logic is how the truth of some propositions is connected
with the truth of another. Thus, we will usually consider a group of related
propositions.
An argument is a set of two or more propositions related to each other in
such a way that all but one of them (the premises) are supposed to provide
support for the remaining one (the conclusion).
The transition from premises to conclusion is the inference upon which the
argument relies.
Example 3.1
Show that the propositions ”The star is made of milk, and strawberries are
red. My dog has ﬂeas.” do not form an argument.
Solution.
Indeed, the truth or falsity of each of the propositions has no bearing on that
of the others
Example 3.2
Show that the propositions:”Mark is a lawyer. So Mark went to law school
since all lawyers have gone to law school” form an argument.
Solution.
This is an argument. The truth of the conclusion, ”Mark went to law school,”
is inferred or deduced from its premises, ”Mark is a lawyer” and ”all lawyers
have gone to law school.”
The above argument can be represented as follows: Let
p: Mark is a lawyer.
q: All lawyers have gone to law school.
r: Mark went to law school.
Then
p∧q
..
˙ r
The symbol .. is to indicate the inferrenced conclusion.
˙
Now, suppose that the premises of an argument are all true. Then the
25.
3 RULES OF INFERENTIAL LOGIC 25
conclusion may be either true or false. When the conclusion is true then the
argument is said to be valid. When the conclusion is false then the argument
is said to be invalid.
To test an argument for validity one proceeds as follows:
(1) Identify the premises and the conclusion of the argument.
(2) Construct a truth table including the premises and the conclusion.
(3) Find rows in which all premises are true.
(4) In each row of Step (3), if the conclusion is true then the argument is
valid; otherwise the argument is invalid.
Example 3.3
Show that the argument
p → q
q → p
.. p ∨ q
˙
is invalid
Solution.
We construct the truth table as follows.
p q p→q q→p p∨q
T T T T T
T F F T T
F T T F T
F F T T F
From the last row we see that the premises are true but the conclusion is
false. The argument is then invalid
Example 3.4 (Modus Ponens or the method of aﬃrming)
a. Show that the argument
p → q
p
.. q
˙
is valid.
b. Show that the argument
∼p∨q → r
∼p∨q
..
˙ r
is valid.
26.
26 FUNDAMENTALS OF MATHEMATICAL LOGIC
Solution.
a. The truth table is as follows.
p q p→q
T T T
T F F
F T T
F F T
The ﬁrst row shows that the argument is valid.
b. Follows from by replacing p with ∼ ∨q and q with r
Example 3.5
Show that the argument
p → q
q
.. p
˙
is invalid.
Solution.
The truth table is as follows.
p q p→q
T T T
T F F
F T T
F F T
Because of the third row the argument is invalid. An argument of this form is
referred to as converse error because the conclusion of the argument would
follows from the premises if p → q is replaced by its converse q → p
Example 3.6 (Modus Tollens or the method of denial)
Show that the argument
p → q
∼q
.. ∼ p
˙
is valid.
27.
3 RULES OF INFERENTIAL LOGIC 27
Solution.
The truth table is as follows.
p q p→q ∼q ∼p
T T T F F
T F F T F
F T T F T
F F T T T
The last row shows that the argument is valid
Example 3.7
Show that the argument
p → q
∼p
.. ∼ q
˙
is invalid.
Solution.
The truth table is as follows.
p q p→q ∼q ∼p
T T T F F
T F F T F
F T T F T
F F T T T
The third row shows that the argument is invalid. This is known as inverse
error because the conclusion of the argument would follow from the premises
if p → q is replaced by the inverse q → p
Example 3.8 (Disjunctive Addition)
a. Show that the argument
p
.. p ∨ q
˙
is valid.
b. Show that the argument
q
.. p ∨ q
˙
is valid.
28.
28 FUNDAMENTALS OF MATHEMATICAL LOGIC
Solution.
a. The truth table is as follows.
p q p∨q
T T T
T F T
F T T
F F F
The ﬁrst and second rows show that the argument is valid.
b. The ﬁrst and third rows show that the argument is valid
Example 3.9 (Conjunctive Simpliﬁcation)
a. Show that the argument
p∧q
..
˙ p
is valid.
b. Show that the argument
p∧q
..
˙ q
is valid.
Solution.
a. The truth table is as follows.
p q p∧q
T T T
T F F
F T F
F F F
The ﬁrst row shows that the argument is valid.
b. The ﬁrst row shows that the argument is valid
Example 3.10 (Disjunctive Syllogism)
a. Show that the argument
p∨q
∼q
..
˙ p
29.
3 RULES OF INFERENTIAL LOGIC 29
is valid.
b. Show that the argument
p∨q
∼p
..
˙ q
is valid.
Solution.
a. The truth table is as follows.
p q ∼p ∼q p∨q
T T F F T
T F F T T
F T T F T
F F T T F
The second row shows that the argument is valid.
b. The third row shows that the argument is valid
Example 3.11 (Hypothetical Syllogism)
Show that the argument
p→q
q→r
..
˙ p→r
is valid.
Solution.
The truth table is as follows.
p q r p→q q→r p→r
T T T T T T
T T F T F F
T F T F T T
T F F F T F
F T T T T T
F T F T F T
F F T T T T
F F F T T T
The ﬁrst , ﬁfth, seventh, and eighth rows show that the argument is valid
30.
30 FUNDAMENTALS OF MATHEMATICAL LOGIC
Example 3.12 (Rule of contradiction)
Show that if c is a contradiction then the following argument is valid for any
p.
∼p→c
..
˙ p
Solution.
Constructing the truth table we ﬁnd
c p ∼p→c
F T T
F F F
The ﬁrst row shows that the argument is valid
31.
3 RULES OF INFERENTIAL LOGIC 31
Review Problems
Problem 3.1
Use modus ponens or modus tollens to ﬁll in the blanks in the argument
below so as to produce valid inferences.
√ √
If 2 is rational, then 2 = a for some integers a and b.
√ b
It is not true that 2 = a for some integers a and b.
b
..
˙
Problem 3.2
Use modus ponens or modus tollens to ﬁll in the blanks in the argument
below so as to produce valid inferences.
If logic is easy, then I am a monkey’s uncle.
I am not a monkey’s uncle.
..
˙
Problem 3.3
Use truth table to determine whether the argument below is valid.
p→q
q→p
.. p ∨ q
˙
Problem 3.4
Use truth table to determine whether the argument below is valid.
p
p→q
∼q∨r
..
˙ r
Problem 3.5
Use symbols to write the logical form of the given argument and then use a
truth table to test the argument for validity.
If Tom is not on team A, then Hua is on team B.
If Hua is not on team B, then Tom is on team A.
.. Tom is not on team A or Hua is not on team B.
˙
32.
32 FUNDAMENTALS OF MATHEMATICAL LOGIC
Problem 3.6
Use symbols to write the logical form of the given argument. If the argument
is valid, identify the rule of inference that guarantees its validity. Otherwise
state whether the converse or the inverse error is made.
If Jules solved this problem correctly, then Jules obtained the answer 2.
Jules obtained the answer 2.
.. Jules solved this problem correctly.
˙
Problem 3.7
Use symbols to write the logical form of the given argument. If the argument
is valid, identify the rule of inference that guarantees its validity. Otherwise
state whether the converse or the inverse error is made.
If this number is larger than 2, then its square is larger than 4.
This number is not larger than 2.
.. The square of this number is not larger than 4.
˙
Problem 3.8
Use the valid argument forms of this section to deduce the conclusion from
the premises.
∼p∨q →r
s∨ ∼ q
∼t
p→t
∼ p ∧ r →∼ s
..
˙ ∼q
Problem 3.9
Use the valid argument forms of this section to deduce the conclusion from
the premises.
∼ p → r∧ ∼ s
t→s
u →∼ p
∼w
u∨w
..
˙ ∼t∨w
33.
4 PROPOSITIONS AND QUANTIFIERS 33
4 Propositions and Quantiﬁers
Statements such as ”x > 3” are often found in mathematical assertions and
in computer programs. These statements are not propositions when the
variables are not speciﬁed. However, one can produce propositions from
such statements.
A predicate is an expression involving one or more variables deﬁned on some
domain, called the domain of discourse. Substitution of a particular value
for the variable(s) produces a proposition which is either true or false. For
instance, P (n) : n is prime is a predicate on the natural numbers. Observe
that P (1) is false, P (2) is true. In the expression P (x), x is called a free
variable. As x varies the truth value of P (x) varies as well. The set of true
values of a predicate P (x) is called the truth set and will be denoted by TP .
Example 4.1
Let Q(x, y) : x = y+3 with domain the collection of natural numbers (i.e. the
numbers 0, 1, 2, · · · ). What are the truth values of the propositions Q(1, 2)
and Q(3, 0)?
Solution.
By substitution in the expression of Q we ﬁnd: Q(1, 2) is false since 1 = x =
y + 3 = 5. On the contrary, Q(3, 0) is true since x = 3 = 0 + 3 = y + 3
If P (x) and Q(x) are two predicates with a common domain D then the
notation P (x) ⇒ Q(x) means that every element in the truth set of P (x) is
also an element in the truth set of Q(x).
Example 4.2
Consider the two predicates P (x) : x is a factor of 4 and Q(x) : x is a factor
of 8. Show that P (x) ⇒ Q(x).
Solution.
Finding the truth set of each predicate we have: TP = {1, 2, 4} and TQ =
{1, 2, 4, 8}. Since every number appearing in TP also appears in TQ then
P (x) ⇒ Q(x)
If two predicates P (x) and Q(x) with a common domain D are such that
TP = TQ then we use the notation P (x) ⇔ Q(x).
34.
34 FUNDAMENTALS OF MATHEMATICAL LOGIC
Example 4.3
Let D = IR. Consider the two predicates P (x) : −2 ≤ x ≤ 2 and Q(x) :
|x| ≤ 2. Show that P (x) ⇔ Q(x).
Solution.
Indeed, if x in TP then the distance from x to the origin is at most 2. That
is, |x| ≤ 2 and hence x belongs to TQ . Now, if x is an element in TQ then
|x| ≤ 2,i.e. (x−2)(x+2) ≤ 0. Solving this inequality we ﬁnd that −2 ≤ x ≤ 2.
That is, x ∈ TP
Another way to generate propositions is by means of quantiﬁers. For exam-
ple ∀x ∈ D, P (x) is a proposition which is true if P (x) is true for all values
of x in the domain D of P. For example, if k is an nonnegative integer, then
the predicate P (k) : 2k is even is true for all k ∈ IN. We write,
∀k ∈ IN, (2k is even).
The symbol ∀ is called the universal quantiﬁer.
The proposition ∀x ∈ D, P (x) is false if P (x) is false for at least one value
of x. In this case x is called a counterexample.
Example 4.4
1
Show that the proposition ∀x ∈ IR, x > x
is false.
Solution.
A counterexample is x = 1 . Clearly,
2
1
2
< 2 = 1.
1
2
Example 4.5
Write in the form ∀x ∈ D, P (x) the proposition :” every real number is either
positive, negative or 0.”
Solution.
∀x ∈ IR, x > 0, x < 0, or x = 0.
The notation ∃x ∈ D, P (x) is a proposition that is true if there is at least
one value of x ∈ D where P (x) is true; otherwise it is false. The symbol ∃ is
called the existential quantiﬁer.
35.
4 PROPOSITIONS AND QUANTIFIERS 35
Example 4.6
Let P (x) denote the statement ”x > 3.” What is the truth value of the
proposition ∃x ∈ IR, P (x).
Solution.
Since 4 ∈ IR and 4 > 3, the given proposition is true
The proposition ∀x ∈ D, P (x) → Q(x) is called the universal conditional
proposition. For example, the proposition ∀x ∈ IR, if x > 2 then x2 > 4 is
a universal conditional proposition.
Example 4.7
Rewrite the proposition ”if a real number is an integer then it is a rational
number” as a universal conditional proposition.
Solution.
∀x ∈ IR, if x is an interger then x is a rational number
Example 4.8
a. What is the negation of the proposition ∀x ∈ D, P (x)?
b. What is the negation of the proposition ∃x ∈ D, P (x)?
c. What is the negation of the proposition ∀x ∈ D, P (x) → Q(x)?
Solution.
a. ∃x ∈ D, ∼ P (x).
b. ∀x ∈ D, ∼ P (x).
c. Since P (x) → Q(x) ≡ (∼ P (x)) ∨ Q(x) then ∼ (∀x ∈ D, P (x) → Q(x)) ≡
∃x ∈ D, P (x) and ∼ Q(x)
Example 4.9
Consider the universal conditional proposition
∀x ∈ D, if P (x) then Q(x).
a. Find the contrapositive.
b. Find the converse.
c. Find the inverse.
36.
36 FUNDAMENTALS OF MATHEMATICAL LOGIC
Solution.
a. ∀x ∈ D, if ∼ Q(x) then ∼ P (x).
b. ∀x ∈ D, if Q(x) then P (x).
c. ∀x ∈ D, if ∼ P (x) then ∼ Q(x)
Example 4.10
Write the negation of each of the following propositions:
a. ∀x ∈ IR, x > 3 → x2 > 9.
b. Every polynomial function is continuous.
c. There exists a triangle with the property that the sum of angles is greater
than 180◦ .
Solution.
a. ∃x ∈ IR, x > 3 and x2 ≤ 9.
b. There exists a polynomial that is not continuous everywhere.
c. For any triangle, the sum of the angles is less than or equal to 180◦
Next, we discuss predicates that contain multiple quantiﬁers. A typical ex-
ample is the deﬁnition of a limit. We say that L = limx→a f (x) if and only if
∀ > 0, ∃ a positive number δ such that if |x − a| ≤ δ then |f (x) − L| < .
Example 4.11
a. Let P (x, y) denote the statement ”x + y = y + x.” What is the truth value
of the proposition (∀x ∈ IR)(∀y ∈ IR), P (x, y)?
b. Let Q(x, y) denote the statement ”x + y = 0.” What is the truth value of
the proposition (∃y ∈ IR)(∀x ∈ IR), Q(x, y)?
Solution.
a. The given proposition is always true.
b. The proposition is false. For otherwise, one can choose x = −y to obtain
0 = x + y = 0 which is impossible
Example 4.12
Find the negation of the following propositions:
a. ∀x∃y, P (x, y).
b. ∃x∀y, P (x, y).
37.
4 PROPOSITIONS AND QUANTIFIERS 37
Solution.
a. ∃x∀y, ∼ P (x, y).
b. ∀x∃y, ∼ P (x, y)
Example 4.13
The symbol ∃! stands for the phrase ”there exists a unique”. Which of the
following statements are true and which are false.
a. ∃!x ∈ IR, ∀y ∈ IR, xy = y.
1
b. ∃! integer x such that x is an integer.
Solution.
a. True. Let x = 1.
b. False since 1 and −1 are both integers with integer reciprocals
38.
38 FUNDAMENTALS OF MATHEMATICAL LOGIC
Review Problems
Problem 4.1
By ﬁnding a counterexample, show that the proposition:” For all positive
integers n and m, m.n ≥ m + n” is false.
Problem 4.2
Consider the statement
∃x ∈ IR such that x2 = 2.
Which of the following are equivalent ways of expressing this statement?
a. The square of each real number is 2.
b. Some real numbers have square 2.
c. The number x has square 2, for some real number x.
d. If x is a real number, then x2 = 2.
e. Some real number has square 2.
f. There is at least one real number whose square is 2.
Problem 4.3
Rewrite the following propositions informally in at least two diﬀerent ways
without using the symbols ∃ and ∀ :
a. ∀ squares x, x is a rectangle.
b. ∃ a set A such that A has 16 subsets.
Problem 4.4
Rewrite each of the following statements in the form ”∃ x such that ”:
a. Some exercises have answers.
b. Some real numbers are rational.
Problem 4.5
Rewrite each of the following statements in the form ”∀ , if then .”:
a. All COBOL programs have at least 20 lines.
b. Any valid argument with true premises has a true conclusion.
c. The sum of any two even integers is even.
d. The product of any two odd integers is odd.
39.
4 PROPOSITIONS AND QUANTIFIERS 39
Problem 4.6
Which of the following is a negation for ”Every polynomial function is con-
tinuous”?
a. No polynomial function is continuous.
b. Some polynomial functions are continuous.
c. Every polynomial function fails to be continuous.
d. There is a noncontinuous polynomial function.
Problem 4.7
Determine whether the proposed negation is correct. If it is not, write a
correct negation.
Proposition : For all integers n, if n2 is even then n is even.
Proposed negation : For all integer n, if n2 is even then n is not even.
Problem 4.8
Let D = {−48, −14, −8, 0, 1, 3, 16, 23, 26, 32, 36}. Determine which of the fol-
lowing propositions are true and which are false. Provide counterexamples
for those propositions that are false.
a. ∀x ∈ D, if x is odd then x > 0.
b. ∀x ∈ D, if x is less than 0 then x is even.
c. ∀x ∈ D, if x is even then x ≤ 0.
d. ∀x ∈ D, if the ones digit of x is 2, then the tens digit is 3 or 4.
e. ∀x ∈ D, if the ones digit of x is 6, then the tens digit is 1 or 2
Problem 4.9
Write the negation of the proposition :∀x ∈ IR, if x(x + 1) > 0 then x > 0 or
x < −1.
Problem 4.10
Write the negation of the proposition : If an integer is divisible by 2, then it
is even.
Problem 4.11
Given the following true propostion:” ∀ real numbers x, ∃ an integer n such
that n > x.” For each x given below, ﬁnd an n to make the predicate n > x
true.
10
a. x = 15.83 b. x = 108 c. x = 1010 .
40.
40 FUNDAMENTALS OF MATHEMATICAL LOGIC
Problem 4.12
Given the proposition: ∀x ∈ IR, ∃ a real number y such that x + y = 0.
a. Rewrite this proposition in English without the use of the quantiﬁers.
b. Find the negation of the given proposition.
Problem 4.13
Given the proposition: ∃x ∈ IR, ∀y ∈ IR, x + y = 0.
a. Rewrite this proposition in English without the use of the quantiﬁers.
b. Find the negation of the given proposition.
Problem 4.14
Consider the proposition ”Somebody is older than everybody.” Rewrite this
proposition in the form ”∃ a person x such that ∀ .”
Problem 4.15
Given the proposition: There exists a program that gives the correct answer
to every question that is posed to it.”
a. Rewrite this proposition using quantiﬁers and variables.
b. Find a negation for the given proposition.
Problem 4.16
Given the proposition: ∀x ∈ IR, ∃y ∈ IR such that x < y.
a. Write a proposition by interchanging the symbols ∀ and ∃.
b. State which is true: the given proposition, the one in part (a), neither, or
both.
Problem 4.17
Find the contrapositive, converse, and inverse of the proposition ”∀x ∈ IR, if
x(x + 1) > 0 then x > 0 or x < −1.”
Problem 4.18
Rewrite the following proposition in if-then form :” Earning a grade of C −
in this course is a suﬃcient condition for it to count toward graduation.”
Problem 4.19
Rewrite the following proposition in if-then form :” Being on time each day
is a necessary condition for keeping this job.”
Problem 4.20
Rewrite the following proposition without using the words ”necessary” or ”
suﬃcient” : ”Divisibility by 4 is not a necessary condition for divisibility by
2.”
41.
5 ARGUMENTS WITH QUANTIFIED PREMISES 41
5 Arguments with Quantiﬁed Premises
In this section we discuss three types of valid arguments that involve the
universal quantiﬁer.
• The rule of universal instantiation:
∀x ∈ D, P (x)
a∈D
..
˙ P (a)
Example 5.1
Use universal instantiation to ﬁll in valid conclusion for the following argu-
ment.
All positive integers are greater than or equal to 1
3 is a positive integer
..
˙
Solution.
All positive integers are greater than or equal to 1
3 is a positive integer
.. 3 ≥ 1
˙
• Universal Modus Ponen:
∀x ∈ D, if P (x) then Q(x)
P (a) f or some a ∈ D
..
˙ Q(a)
Example 5.2
Use the rule of the universal modus ponens to ﬁll in valid conclusion for the
following argument.
∀n ∈ IN, if n = 2k for some k ∈ IN then n is even.
0 = 2.0
..
˙
Solution.
∀n ∈ IN, if n = 2k for some k ∈ IN then n is even.
42.
42 FUNDAMENTALS OF MATHEMATICAL LOGIC
0 = 2.0
..0 is even
˙
• Universal Modus Tollens:
∀x ∈ D, if P (x) then Q(x)
∼ Q(a) f or some a ∈ D
..
˙ ∼ P (a)
Example 5.3
Use the rule of the universal modus tonens to ﬁll in valid conclusion for the
following argument.
All healthy people eat an apple a day.
Harry does not eat an apple a day.
..
˙
Solution.
All healthy people eat an apple a day.
Harry does not eat an apple a day.
.. Harry is not healthy
˙
Next, we discuss a couple of invalid arguments whose premises involve quan-
tiﬁers.
• The rule of converse error:
∀x ∈ D, if P (x) then Q(x)
Q(a) f or some a ∈ D
..
˙ P (a)
Example 5.4
What kind of error does the following invalid argument exhibit?
All healthy people eat an apple a day.
Helen eats an apple a day.
.. Helen is healthy
˙
Solution.
This invalid argument exhibits the converse error
43.
5 ARGUMENTS WITH QUANTIFIED PREMISES 43
• The rule of inverse error:
∀x ∈ D, if P (x) then Q(x)
∼ P (a) f or some a ∈ D
..
˙ ∼ Q(a)
Example 5.5
What kind of error does the following invalid argument exhibit?
All healthy people eat an apple a day.
Hubert is not a healthy person.
.. Hubert does not eat an apple a day.
˙
Solution.
This invalid argument exhibits the inverse error
44.
44 FUNDAMENTALS OF MATHEMATICAL LOGIC
Review Problems
Problem 5.1
Use the rule of universal modus ponens to ﬁll in valid conclusion for the ar-
gument.
For all real numbers a, b, c, and d, if b = 0 and d = 0 then a + d = ad+bc .
b
c
bd
a = 2, b = 3, c = 4, and d = 5 are particular real numbers such that b = 0
and d = 0.
..
˙
Problem 5.2
Use the rule of universal modus tonens to ﬁll in valid conclusion for the ar-
gument.
If a computer is correct, then compilation of the program does not produce
error messages.
Compilation of this program produces error messages.
..
˙
Problem 5.3
Use the rule of universal modus ponens to ﬁll in valid conclusion for the ar-
gument.
All freshmen must take writing.
Caroline is a freshman.
..
˙ .
Problem 5.4
What kind of error does the following invalid argument exhibit?
All cheaters sit in the back row.
George sits in the back row.
.. George is a cheater.
˙
Problem 5.5
What kind of error does the following invalid argument exhibit?
All honest people pay their taxes.
Darth is not honest.
.. Darth does not pay his taxes.
˙
45.
6 PROJECT I: DIGITAL LOGIC DESIGN 45
6 Project I: Digital Logic Design
In this section we discuss the logic of digital circuits which are considered to
be the basic components of most digital systems, such as electronic comput-
ers, electronic phones, traﬃc light controls, etc.
The purpose of digital systems is to manipulate discrete information which
are represented by physical quantities such as voltages and current. The
smallest representation unit is one bit, short for binary digit. Since electronic
switches have two physical states, namely high voltage and low voltage we
attribute the bit 1 to high voltage and the bit 0 for low voltage.
A logic gate is the smallest processing unit in a digital system. It takes one
or few bits as input and generates one bit as an output.
A circuit is composed of a number of logic gates connected by wires. It
takes a group of bits as input and generates one or more bits as output.
The six basic logic gates are the following:
(1) NOT gate (also called inverter): Takes an input of 0 to an output
of 1 and an input of 1 to an output of 0. The corresponding logical symbol
is ∼ P.
(2) AND gate: Takes two bits, P and Q, and outputs 1 if P and Q are 1 and
0 otherwise. The logical symbol is P ∧ Q.
(3) OR gate: outputs 1 if either P or Q is 1 and 0 otherwise. The logical
symbol is P ∨ Q.
(4) NAND gate: outputs a 0 if both P and Q are 1 and 1 otherwise. The
symbol is ∼ (P ∧ Q). Also, denoted by P |Q, where | is called a Scheﬀer
stroke.
(5) NOR gate: output a 0 if at least one of P or Q is 1 and 1 otherwise. The
symbol is ∼ (P ∨ Q) or P ↓ Q, where ↓ is a Pierce arrow
The logic gates have the following graphical representations:
46.
46 FUNDAMENTALS OF MATHEMATICAL LOGIC
Problem 6.1
Construct the truth tables of the gates discussed in this section.
If you are given a set of input signals for a circuit, you can ﬁnd its output
by tracing through the circuit gate by gate.
Problem 6.2
Give the output signal S for the following circuit, given that P = 0, Q = 1,
and R = 0 :
Problem 6.3
Write the input/output table for the circuit of the previous problem.
A variable with exactly two possible values is called a Boolean variable.
A Boolean expression is an expression composed of Boolean variables and
connectives (which are the gates in this section).
47.
6 PROJECT I: DIGITAL LOGIC DESIGN 47
Problem 6.4
Find the Boolean expression that corresponds to the circuit of Problem 6.1.
Problem 6.5
Construct the circuit corresponding to the Boolean expression: (P ∧ Q)∨ ∼
R.
Problem 6.6
For the following input/output table, construct (a) the corresponding Boolean
expression and (b) the corresponding circuit:
P Q R S
1 1 1 0
1 1 0 1
1 0 1 0
1 0 0 0
0 1 1 1
0 1 0 0
0 0 1 0
0 0 0 0
Two digital logic circuits are equivalent if, and only if, their corresponding
Boolean expressions are logically equivalent.
Problem 6.7
Show that the following two circuits are equivalent:
Problem 6.8
Consider the following circuit
48.
48 FUNDAMENTALS OF MATHEMATICAL LOGIC
Let P and Q be single binary digits and P + Q = RS. Complete the fol-
lowing table
P Q R S
1 1
1 0
0 1
0 0
The given circuit is called a half-adder. It computes the sum of two single
binary digits.
Several methods have been used for expressing negative numbers in the com-
puter. The most obvious way is to convert the number to binary and stick
on another bit to indicate sign, 0 for positive and 1 for negative. Suppose
that integers are stored using this signed-magnitude technique in 8 bits so
that the leftmost bit holds the sign while the remaining bits represent the
magnitude. Thus, +4110 = 00101001 and −4110 = 10101001.
The above procedure has a gap. How one would represent the bit 0? Well,
there are two ways for storing 0. One way is 00000000 which represents
+0 and a second way 10000000 represents −0. A method for representing
numbers that avoid this problem is called the two’s complement. Con-
sidering −4110 again, ﬁrst, convert the absolute value to binary obtaining
4110 = 00101001. Then take the complement of each bit obtaining 11010110.
This is called the one complement of 41. To complete the procedure, in-
crement by 1 the one’s complement to obtain −4110 = 11010111.
Conversion of +4110 to two’s complement consists merely of expressing the
number in binary, i.e. +4110 = 00101001.
49.
6 PROJECT I: DIGITAL LOGIC DESIGN 49
Problem 6.9
Express the numbers 104 and −104 in two’s complement representation with
8 bits.
Now, an algorithm to ﬁnd the decimal representation of the integer with a
given 8-bit two’s complement is the following:
1. Find the two’s complement of the given two’s complement,
2. write the decimal equivalent of the result.
Problem 6.10
What is the decimal representation for the integer with two’s complement
10101001?
50.
50 FUNDAMENTALS OF MATHEMATICAL LOGIC
7 Project II: Number Systems
In this section we consider three number systems that are of importance
in applications, namely, the decimal system, the binary system, and the
hexadecimal system. Decimal numbers are used in communication among
human beings whereas binary numbers are used by computers to represent
numbers.
Consider ﬁrst the decimal system. If n is a positive integer then n can
be written as
n = dk dk−1 · · · d1 d0 ,
where the digits d0 , d1 , · · · , dk are elements of the set {0, 1, 2, · · · , 9}.
The number n can be expressed as a sum of powers of 10 as follows:
n = dk 10k + dk−1 10k−1 + · · · + d1 101 + d0 100 .
For example,
5049 = 5(103 ) + 0(102 ) + 4(101 ) + 9(100 ).
A number in binary system is a number n that can be written in the form
n = bk bk−1 · · · b1 b0 ,
where bi is either 0 or 1.
We will use subscript to tell the base of which a number is represented.
Thus, we write n2 = bk bk−1 · · · b1 b0 to indicate that the number n is in base
2.
If n is a number in base 2 then its decimal value (i.e. base 10) is found by
the formula:
n2 = bk (2k ) + bk−1 (2k−1 ) + · · · + b1 (21 ) + b0 (20 ) = m10 .
Problem 7.1
Find the decimal value of the following binary numbers:
a. 11001012
b. 1101102
51.
7 PROJECT II: NUMBER SYSTEMS 51
To convert a positive integer n from base 10 to base 2 we use the division
algorithm as follows:
(1) n = q0 (2) + r0 , where q0 is the quotient of the division of n by 2 and r0
is the remainder.
(2) If q0 = 0 then n is already in base 2. If not then divide q0 by 2 to obtain
q0 = q1 (2) + r1 .
(3) If q1 = 0 then n10 = r1 r0 . If not repeat the process. Note that the
remainders are all less than 2.
Suppose that qk = 0 then
n10 = rk rk−1 · · · r1 r0 .
Problem 7.2
Represent the following decimal integers in binary notation:
a. 129710
b. 45810
Problem 7.3
Evaluate the following sums:
a. 110111012 + 10010110102
b. 1011012 + 111012
Another useful number system is the hexadecimal system. The possible
digits in an hexadecimal system are :
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
where A, B, C, D, E, F stand for 10, 11, 12, 13, 14, and 15 respectively.
The conversion of a number from base 16 to base 10 is similar to the conver-
sion of numbers from base 2 to base 10. The conversion of a number from
base 10 to base 16 is similar to the conversion of a decimal number to base
2.
Problem 7.4
Convert the number A2BC16 to base 10.
To convert an integer from base 16 to base 2 one performs the following:
(1) Write each hexadecimal digit of the integer in ﬁxed 4-bit binary notation.
(2) Juxtapose the results.
52.
52 FUNDAMENTALS OF MATHEMATICAL LOGIC
Problem 7.5
Convert the number B53DF 816 to base 2.
To convert an integer from base 2 to base 16:
(1) Group the digits of the binary number into sets of four bits, starting from
the right and adding leading zeros as needed.
(2) Convert the binary numbers in (1) to base 16.
(3) Juxtapose the results of (2)
Problem 7.6
Convert the number 1011011110001012 to base 16.
53.
Fundamentals of Mathematical
Proofs
In this chapter we discuss some common methods of proof and the standard
terminology that accompanies them.
8 Methods of Direct Proof I
A mathematical system consists of axioms, deﬁnitions, and undeﬁned
terms. An axiom is a statement that is assumed to be true. A deﬁnition
is used to create new concepts in terms of existing ones. A theorem is a
proposition that has been proved to be true. A lemma is a theorem that
is usually not interesting in its own right but is useful in proving another
theorem. A corollary is a theorem that follows quickly from a theorem.
Example 8.1
The Euclidean geometry furnishes an example of mathematical system:
• points and lines are examples of undeﬁned terms.
• An example of a deﬁnition: Two angles are supplementary if the sum of
their measures is 180◦ .
• An example of an axiom: Given two distinct points, there is exactly one
line that contains them.
• An example of a theorem: If two sides of a triangle are equal, then the
angles opposite them are equal.
• An example of a corollary: If a triangle is equilateral, then it is equiangular.
An argument that establishes the truth of a theorem is called a proof. Logic
is a tool for the analysis of proofs.
53
54.
54 FUNDAMENTALS OF MATHEMATICAL PROOFS
First we discuss methods for proving a theorem of the form ”∃x such that
P (x).” This theorem guarantees the existence of at least one x for which the
predicate P (x) is true. The proof of such a theorem is constructive: that
is, the proof is either by ﬁnding a particular x that makes P (x) true or by
exhibiting an algorithm for ﬁnding x.
Example 8.2
Show that there exists a positive integer which can be written as the sum of
the squares of two numbers.
Solution.
Indeed, one example is 52 = 32 + 42
Example 8.3
Show that there exists an integer x such that x2 = 15, 129.
Solution.
Applying the well-known algorithm of extracting the square root we ﬁnd that
x = 123
By a nonconstructive existence proof we mean a method that involves
either showing the existence of x using a proved theorem (or axioms) or the
assumption that there is no such x leads to a contradiction. The disadvan-
tage of nonconstructive method is that it may give virtually no clue about
where or how to ﬁnd x.
Theorems are often of the form ”∀x ∈ D if P (x) then Q(x).” We call P (x)
the hypothesis and Q(x) the conclusion.
Let us ﬁrst consider a proposition of the form ∀x ∈ D, P (x). Then this
can be written in the form ”∀x, if x ∈ D then P (x).” If D is a ﬁnite set, then
one check the truth value of P (x) for each x ∈ D. This method is called the
method of exhaustion.
Example 8.4
Show that for each integer 1 ≤ n ≤ 10, n2 − n + 11 is a prime number.
Solution.
The given proposition can be written in the form ”∀n ∈ IN, if 1 ≤ n ≤ 10
55.
8 METHODS OF DIRECT PROOF I 55
then P (n)” where P (n) = n2 − n + 11. Using the method of exhaustion we
see that
P (1) = 11 ; P (2) = 13 ; P (3) = 17 ; P (4) = 23
P (5) = 31 ; P (6) = 41 ; P (7) = 53 ; P (8) = 67
P (9) = 83 ; P (10) = 101.
The most powerful technique for proving a universal proposition is one that
works regardless of the size of the domain over which the proposition is
quantiﬁed. It is called the method of generalizing from the generic
particular.
The method consists of picking an arbitrary element x of the domain (known
as a generic element) for which the hypothesis P (x) is satisﬁed, and then
using deﬁnitions, previously established results, and the rules of inference to
conclude that Q(x) is also true.
By a direct method of proof we mean a method that consists of showing
that if P (x) is true for x ∈ D then Q(x) is also true.
The following shows the format of the proof of a theorem.
Theorem 8.1
For all n, m ∈ Z , if m and n are even then so is m + n.
Proof.
Let m and n be two even integers. Then there exist integers k1 and k2 such
that n = 2k1 and m = 2k2 . We must show that m + n is even, that is, an
integer multiple of 2. Indeed,
m + n = 2k1 + 2k2
= 2(k1 + k2 )
= 2k
where k = k1 + k2 ∈ Z . Thus, by the deﬁnition of even, m + n is even
Example 8.5
Prove the following theorem.
Theorem Every integer is a rational number.
56.
56 FUNDAMENTALS OF MATHEMATICAL PROOFS
Solution.
n
Proof. Let n be an arbitrary integer. Then n = 1
. By the deﬁnition of
rational numbers, n is rational
Theorem 8.2
If a, b ∈ Q then a + b ∈ Q.
I I
Proof.
Let a and b be two rational numbers. Then there exist integers a1 , a2 , b1 = 0,
and b2 = 0 such that a = a1 and b = a2 . By the property of addition of two
b
1
b
2
fractions we have
a + b = a1 + a2
b
1
b
2
a1 b2 +a2 b1
= b1 b2
By letting p = a1 b2 + a2 b1 ∈ Z and q = b1 b2 ∈ Z ∗ we get a + b = p . That is,
q
a + b ∈QI
Corollary 8.1 The double of a rational number is rational.
Proof.
Let a = b in the previous theorem we see that 2a = a + a = a + b ∈ Q
I
Next, we point out of some common mistakes that must be avoided in prov-
ing theorems.
• Arguing from examples. The validity of a general statement can not be
proved by just using a particular example.
• Using the same letters to mean two diﬀerent things. For example, sup-
pose that m and n are any two given even integers. Then by writing m = 2k
and n = 2k this would imply that m = n which is inconsistent with the
statement that m and n are arbitrary.
• Jumping to a conclusion. Let us illustrate by an example. Suppose that
we want to show that if the sum of two integers is even so is their diﬀerence.
Consider the following proof: Suppose that m + n is even. Then there is an
integer k such that m + n = 2k. Then, m = 2k − n and so m − n is even.
The problem with this proof is that the crucial step m − n = 2k − n − n =
2(k − n) is missing. The author of the proof has jumped prematurely to a
57.
8 METHODS OF DIRECT PROOF I 57
conclusion.
• Begging the question. By that we mean that the author of a proof uses in
his argument a fact that he is supposed to prove.
Finally, to show that a proposition of the form ∀x ∈ D, if P (x) then Q(x)
is false it suﬃces to ﬁnd an element x ∈ D where P (x) is true but Q(x) is
false. Such an x is called a counterexample.
Example 8.6
Disprove the proposition ∀a, b ∈ IR, if a < b then a2 < b2 .
Solution.
A counterexample is the following. Let a = −2 and b = −1. Then a < b but
a2 > b 2
58.
58 FUNDAMENTALS OF MATHEMATICAL PROOFS
Review Problems
A real number r is called rational if there exist two integers a and b = 0
such that r = a . A real number that is not rational is called irrational.
b
Problem 8.1
Show that the number r = 6.321521521... is a rational number.
Problem 8.2
Prove the following theorem.
Theorem. The product of two rational numbers is a rational number.
Problem 8.3
Use the previous exercise to prove the following.
Corollary. The square of any rational number is rational.
Problem 8.4
Use the method of constructive proof to show that if r and s are two real
numbers then there exists a real number x such that r < x < s.
Problem 8.5
The following Pascal program segment does not ﬁnd the minimum value in
a data set of N integers. Find a counterexample.
MINN := 0;
FOR I := 1 TO N DO
BEGIN
READLN (A);
If A < MINN THEN MINN := A
END
59.
9 MORE METHODS OF PROOF 59
9 More Methods of Proof
A vacuous proof is a proof of an implication p → q in which it is shown
that p is false.
Example 9.1
Use the method of vacuous proof to show that if x ∈ ∅ then David is playing
pool.
Solution.
Since the proposition x ∈ ∅ is always false, the given proposition is vacuously
true
A trivial proof of an implication p → q is one in which q is shown to
be true without any reference to p.
Example 9.2
Use the method of trivial proof to show that if n is an even integer then n is
divisible by 1.
Solution.
Since the proposition n is divisible by 1 is always true, the given implication
is trivially true
The method of proof by cases is a direct method of proving the condi-
tional proposition p1 ∨ p2 ∨ · · · ∨ pn → q. The method consists of proving the
conditional propositions p1 → q, p2 → q, · · · , pn → q.
Example 9.3
Show that if n is a positive integer then n3 + n is even.
Solution.
We use the method of proof by cases.
Case 1. Suppose that n is even. Then there is k ∈ IN such that n = 2k. In
this case, n3 + n = 8k 3 + 2k = 2(4k 3 + k) which is even.
Case 2. Suppose that n is odd. Then there is a k ∈ IN such that n = 2k + 1.
So, n3 + n = 2(4k 3 + 6k 2 + 4k + 1) which is even
60.
60 FUNDAMENTALS OF MATHEMATICAL PROOFS
Example 9.4
Use the proof by cases to prove the triangle inequality: |x + y| ≤ |x| + |y|.
Solution.
Case 1. x ≥ 0 and y ≥ 0. Then x + y ≥ 0 and so |x + y| = x + y = |x| + |y|.
Case 2. x ≥ 0 and y < 0. Then x + y < x + 0 < |x| ≤ |x| + |y|. On the
other hand, −(x + y) = −x + (−y) ≤ 0 + (−y) = |y| ≤ |x| + |y|. Thus,
if |x + y| = x + y then |x + y| < |x| + |y| and if |x + y| = −(x + y) then
|x + y| ≤ |x| + |y|.
Case 3. The case x < 0 and y ≥ 0 is similar to case 2.
Case 4. Suppose x < 0 and y < 0. Then x + y < 0 and therefore |x + y| =
−(x + y) = (−x) + (−y) = |x| + |y|.
So in all four cases |x + y| ≤ |x| + |y|.
Now, given a real number x. The largest integer n such that n ≤ x < n + 1
is called the ﬂoor of x and is denoted by x . The smallest integer n such
that n − 1 < x ≤ n is called the ceiling of x and is denoted by x .
Example 9.5
Compute x and x of the following values of x :
a. 37.999 b. − 57 c. −14.001
2
Solution.
a. 37.999 = 37, 37.999 = 38.
b. − 57 = −29, − 57 = −28.
2 2
c. −14.001 = −15, −14.001 = −14.
Example 9.6
Use the proof by a counterexample to show that the proposition ”∀x, y ∈
IR, x + y = x + y ” is false.
Solution.
Let x = y = 0.5. Then x + y = 1 and x + y = 0
The following gives another example of the method of proof by cases.
Theorem 9.1
For any integer n,
n n
2
, if n is even
= n−1
2 2
, if n is odd
61.
9 MORE METHODS OF PROOF 61
Proof.
Let n be any integer. Then we consider the following two cases.
Case 1. n is odd. In this case, there is an integer k such that n = 2k + 1.
Hence,
n 2k + 1 1
= = k+ =k
2 2 2
since k ≤ k + 1 < k + 1. Since n = 2k + 1 then solving this equation for k
2
we ﬁnd k = n−1 . It follows that
2
n n−1
=k= .
2 2
Case 2. Suppose n is even. Then there is an integer k such that n = 2k.
Hence, n = k = k = n .
2 2
62.
62 FUNDAMENTALS OF MATHEMATICAL PROOFS
Review Problems
Problem 9.1
Prove that for any integer n the product n(n + 1) is even.
Problem 9.2
Prove that the square of any integer has the form 4k or 4k + 1 for some
integer k
Problem 9.3
Prove that for any integer n, n(n2 − 1)(n + 2) is divisible by 4.
Theorem 9.2
Given any nonnegative integer n and a positive integer d there exist integers
q and r such that n = dq + r and 0 ≤ r < d. The number q is called the
quotient of the division of n by d and we write q = n div d. The number r
is called the remainder and we write r = n mod d or n ≡ r(mod d).
Proof.
The proof uses the fact that any nonempty subset of IN has a smallest ele-
ment. So let S = {n − d · k ∈ IN : k ∈ Z }. This set is nonempty. Indeed, if
n ∈ IN then n = n−0·d ≥ 0 and if n < 0 then n−d·n = n·(1−d) ≥ 0. Thus,
S is a nonempty subset of IN so it has a smallest elements, called r. That is,
there is an integer q such that n − d · q = r or n = d · q + r. It remains to show
that r < d. Suppose the contrary, i.e. r ≥ d. Then n − d · (q + 1) = r − d ≥ 0
so that n − d · (q + 1) ∈ S. Hence, r ≤ n − d · (q + 1) = r − d, a contradiction.
Hence, r < d
The following theorem shows a way for ﬁnding q and r.
Theorem 9.3
n
If n is a nonnegative integer and d is a positive integer by letting q = d
and r = n − d n , we have
d
n = dq + r, and 0 ≤ r < d.
Proof.
n
Suppose n is a nonnegative integer, d is a positive integer,q = d
and
r = n − d n . By substitution we have
d
n n
dq + r = d +n−d = n.
d d
63.
9 MORE METHODS OF PROOF 63
It remains to show that 0 ≤ r < d. By the deﬁnition of the ﬂoor function we
have
n
q ≤ < q + 1.
d
Multiplying through by d we ﬁnd
dq ≤ n < dq + d.
This implies that
0 ≤ n − dq < d.
But
n
r =n−d = n − dq.
d
Hence, 0 ≤ r < d. This completes a proof of the theorem
Problem 9.4
State a necessary and suﬃcient condition for the ﬂoor function of a real
number to equal that number
Problem 9.5
n
Prove that if n is an even integer then 2
= n.
2
Problem 9.6
Show that the equality x − y = x − y is not valid for all real numbers
x and y.
Problem 9.7
Show that the equality x + y = x + y is not valid for all real numbers
x and y.
Problem 9.8
Prove that for all real numbers x and all integers m, x + m = x + m.
Problem 9.9
n n+1
Show that if n is an odd integer then 2
= 2
.
64.
64 FUNDAMENTALS OF MATHEMATICAL PROOFS
10 Methods of Indirect Proofs: Contradiction
and Contraposition
Recall that in a direct proof one starts with the hypothesis of an implication
p → q and then prove that the conclusion is true. Any other method of
proof will be referred to as an indirect proof. In this section we study two
methods of indirect proofs, namely, the proof by contradiction and the proof
by contrapositive.
• Proof by contradiction: We want to show that p is true. we assume it
is not and therefore ∼ p is true and then derive a contradiction. By the rule
of contradiction discussed in Chapter 1, p must be true.
Theorem 10.1
If n2 is an even integer so is n.
Proof.
Suppose the contrary. That is suppose that n is odd. Then there is an integer
k such that n = 2k + 1. In this case, n2 = 2(2k 2 + 2k) + 1 is odd and this
contradicts the assumption that n2 is even. Hence, n must be even
Theorem 10.2
√
The number 2 is irrational.
Proof. √
Suppose not. That is, suppose that 2 is rational. Then there exist two
√
integers m and n with no common divisors such that 2 = m . Squaring
n
both sides of this equality we ﬁnd that 2n2 = m2 . Thus, m2 is even. By
Theorem 10.1, m is even. That is, 2 divides m. But then m = 2k for some
integer k. Taking the square we ﬁnd that 2n2 = m2 = 4k 2 , that is n2 = 2k 2 .
This says that n2 is even and by Theorem 10.1, n is even. We conclude that
2 divides both m and n and this contradcits our assumption that m and n
√
have no common divisors. Hence, 2 must be irrational
Theorem 10.3
The set of prime numbers is inﬁnite.
65.
10 METHODS OF INDIRECT PROOFS: CONTRADICTION AND CONTRAPOSITION65
Proof.
Suppose not. That is, suppose that the set of prime numbers is ﬁnite. Then
these prime numbers can be listed, say, p1 , p2 , · · · , pn . Now, consider the inte-
ger N = p1 p2 · · · pn +1. By the Unique Factorization Theorem, ( See Exercise
??) N can be factored into primes. Thus, there is a prime number pi such
that pi |N. But since pi |p1 p2 · · · pn then pi |(N − p1 p2 · · · pn ) = 1, a contradic-
tion since pi > 1.
• Proof by contrapositive: We already know that p → q ≡∼ q →∼ p. So
to prove p → q we sometimes instead prove ∼ q →∼ p.
Theorem 10.4
If n is an integer such that n2 is odd then n is also odd.
Proof.
Suppose that n is an integer that is even. Then there exists an integer k such
that n = 2k. But then n2 = 2(2k 2 ) which is even.
66.
66 FUNDAMENTALS OF MATHEMATICAL PROOFS
Review Problems
Problem 10.1
Use the proof by contradiction to prove the proposition ”There is no greatest
even integer.”
Problem 10.2
Prove by contradiction that the diﬀerence of any rational number and any
irrational number is irrational.
Problem 10.3
Use the proof by contraposition to show that if a product of two positive real
numbers is greater than 100, then at least one of the numbers is greater than
10.
Problem 10.4
Use the proof by contradiction to show that the product of any nonzero
rational number and any irrational number is irrational.
67.
11 METHOD OF PROOF BY INDUCTION 67
11 Method of Proof by Induction
With the emphasis on structured programming has come the development of
an area called program veriﬁcation, which means your program is correct
as you are writing it.
One technique essential to program veriﬁcation is mathematical induc-
tion, a method of proof that has been useful in every area of mathematics
as well.
Consider an arbitrary loop in Pascal starting with the statement
F OR I := 1 T O N DO
If you want to verify that the loop does something regardless of the particular
integral value of N, you need mathematical induction.
Also, sums of the form
n
n(n + 1)
k=
k=1
2
are very useful in analysis of algorithms and a proof of this formula is math-
ematical induction.
Next we examine this method. We want to prove that some predicate P (n)
is true for any nonnegative integer n ≥ n0 . The steps of mathematical induc-
tion are as follows:
(i) (Basis of induction) Show that P (n0 ) is true.
(ii) (Induction hypothesis) Assume P (n) is true.
(iii) (Induction step) Show that P (n + 1) is true.
Example 11.1
Use the technique of mathematical induction to show that
n(n + 1)
1 + 2 + 3 + ··· + n = , n ≥ 1.
2
Solution.
Let S(n) = 1 + 2 + · · · + n. Then
(i) (Basis of induction) S(1) = 1 = 1(1+1) . That is, S(1)is true.
2
n(n+1)
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) = 2
.
68.
68 FUNDAMENTALS OF MATHEMATICAL PROOFS
(n+1)(n+2)
(iii) (Induction step) We must show that S(n + 1) = 2
. Indeed,
S(n + 1) = 1 + 2 + · · · + n + (n + 1)
= S(n) + (n + 1)
n(n+1)
= 2
+ (n + 1)
(n+1)(n+2)
= 2
Example 11.2 (Geometric progression)
a. Let S(n) = n ark , n ≥ 0 where r = 1. Use induction to show that
k=0
n+1
S(n) = a(1−r ) .
1−r
b. Show that 1 + 1 + · · · + 2n−1 ≤ 2, for all n ≥ 1.
2
1
Solution.
a. We use the method of proof by mathematical induction.
(i) (Basis of induction) S(0) = a = 0 ark . That is, S(1)is true.
k=0
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) = n ark .
k=0
a(1−r n+2 )
(iii) (Induction step) We must show that S(n + 1) = 1−r . Indeed,
n+1 k
S(n + 1) = k=0 ar
n+1
= S(n) + ar
1−r n+1 1−r
= a 1−r + arn+1 1−r
n+1 +r n+1 −r n+2
= a 1−r 1−r
n+2
= a 1−r .
1−r
b. By a. we have
1−( 1 )n
1+ 1 +
2
1
22
+ ··· + 1
2n−1
= 2
1− 12
= 2(1 − ( 1 )n )
2
1
= 2 − 2n−1
≤ 2.
Example 11.3 (Arithmetic progression)
Let S(n) = n (a + (k − 1)r), n ≥ 1. Use induction to show that S(n) =
k=1
n
2
[2a + (n − 1)r].
Solution.
We use the method of proof by mathematical induction.
69.
11 METHOD OF PROOF BY INDUCTION 69
(i) (Basis of induction) S(1) = a = 1 [2a + (1 − 1)r]. That is, S(1)is true.
2
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) = n [2a + (n −
2
1)r].
(iii) (Induction step) We must show that S(n + 1) = (n+1) [2a + nr]. Indeed,
2
n+1
S(n + 1) = k=1 (a + (k − 1)r)
= S(n) + a + (n + 1 − 1)r
n
= 2
[2a + (n − 1)r] + a + nr
2an+n2 r−nr+2a+2nr
= 2
2a(n+1)+n(n+1)r
= 2
n+1
= 2
[2a + nr].
We next exhibit a theorem whose proof uses mathematical induction.
Theorem 11.1
For all integers n ≥ 1, 22n − 1 is divisible by 3.
Proof.
Let P (n) : 22n − 1 is divisible by 3. Then
(i) (Basis of induction) P (1) is true since 3 is divisible by 3.
(ii) (Induction hypothesis) Assume P (n) is true. That is, 22n − 1 is divisible
by 3.
(iii) (Induction step) We must show that 22n+2 − 1 is divisible by 3. Indeed,
22n+2 − 1 = 22n (4) − 1
2n
= 2 (3 + 1) − 1
= 2 · 3 + (22n − 1)
2n
= 22n · 3 + P (n)
Since 3|(22n − 1) and 3|(22n · 3) we have 3|(22n · 3 + 22n − 1). This ends a proof
of the theorem
Example 11.4
a. Use induction to prove that n < 2n for all non-negative integers n.
b. Use induction to prove that 2n < n! for all non-negative integers n ≥ 4.
Solution.
a. Let S(n) = 2n − n, n ≥ 0. We want to show that S(n) > 0 is valid for all
70.
70 FUNDAMENTALS OF MATHEMATICAL PROOFS
n ≥ 0. By the method of mathematical induction we have
(i) (Basis of induction) S(0) = 20 − 0 = 1 > 0. That is, S(0)is true.
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) > 0.
(iii) (Induction step) We must show that S(n + 1) > 0. Indeed,
S(n + 1) = 2n+1 − (n + 1)
= 2n .2 − n − 1
n
= 2 (1 + 1) − n − 1
= 2n − n + 2n − 1
= (2n − 1) + S(n)
> 2n − 1
≥ 0
since the smallest value of n is 0 and in this case 20 − 1 = 0.
b. Let S(n) = n! − 2n , n ≥ 4. We want to show that S(n) > 0 for all n ≥ 4.
By the method of mathematical induction we have
(i) (Basis of induction) S(4) = 4! − 24 = 8 > 0. That is, S(4)is true.
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) > 0, n ≥ 4.
(iii) (Induction step) We must show that S(n + 1) > 0. Indeed,
S(n + 1) = (n + 1)! − 2n+1
= (n + 1)n! − 2n (1 + 1)
= n! − 2n + nn! − 2n
> 2(n! − 2n ) = 2S(n)
> 0
where we have used the fact that if n ≥ 1 then nn! ≥ n!
Example 11.5 (Bernoulli’s inequality)
Let h > −1. Use induction to show that
(1 + nh) ≤ (1 + h)n , n ≥ 0.
Solution.
Let S(n) = (1 + h)n − (1 + nh). We want to show that S(n) ≥ 0 for all n ≥ 0.
We use mathematical induction as follows.
(i) (Basis of induction) S(0) = (1 + h)0 − (1 + 0h) = 0. That is, S(0)is true.
(ii) (Induction hypothesis) Assume S(n) is true. That is, S(n) ≥ 0, n ≥ 0.
71.
11 METHOD OF PROOF BY INDUCTION 71
(iii) (Induction step) We must show that S(n + 1) ≥ 0. Indeed,
S(n + 1) = (1 + h)n+1 − (1 + (n + 1)h)
= (1 + h)(1 + h)n − nh − 1 − h
≥ (1 + h)(1 + nh) − nh − 1 − h
= nh2
≥ 0.
Example 11.6
Deﬁne the following sequence of numbers: a1 = 2 and for n ≥ 2, an = 5an−1 .
Find a formula for an and then prove its validity by mathematical induction.
Solution.
Listing the ﬁrst few terms we ﬁnd, a1 = 2, a2 = 10, a3 = 50, a4 = 250. Thus,
an = 2.5n−1 . We will show that this formula is valid for all n ≥ 1 by the
method of mathematical induction.
(i) (Basis of induction) a1 = 2 = 2.51−1 . That is, a1 is true.
(ii) (Induction hypothesis) Assume an is true. That is, an = 2.5n−1
(iii) (Induction step) We must show that an+1 = 2.5n . Indeed,
an+1 = 5an
= 5(2.5n−1 )
= 2.5n .
72.
72 FUNDAMENTALS OF MATHEMATICAL PROOFS
Review Problems
Problem 11.1
Use the method of induction to show that
2 + 4 + 6 + · · · + 2n = n2 + n
for all integers n ≥ 1.
Problem 11.2
Use mathematical induction to prove that
1 + 2 + 22 + · · · + 2n = 2n+1 − 1
for all integers n ≥ 0.
Problem 11.3
Use mathematical induction to show that
n(n + 1)(2n + 1)
12 + 22 + · · · + n2 =
6
for all integers n ≥ 1.
Problem 11.4
Use mathematical induction to show that
2
3 3 3 n(n + 1)
1 + 2 + ··· + n =
2
for all integers n ≥ 1.
Problem 11.5
Use mathematical induction to show that
1 1 1 n
+ + ··· + =
1·2 2·3 n(n + 1) n+1
for all integers n ≥ 1.
73.
11 METHOD OF PROOF BY INDUCTION 73
Problem 11.6
Use the formula
n(n + 1)
1 + 2 + ··· + n =
2
to ﬁnd the value of the sum
3 + 4 + · · · + 1, 000.
Problem 11.7
Find the value of the geometric sum
1 1 1
1 + + 2 + ··· + n.
2 2 2
Problem 11.8
Let S(n) = n (k+1)! . Evaluate S(1), S(2), S(3), S(4), and S(5). Make a
k=1
k
conjecture about a formula for this sum for general n, and prove your con-
jecture by mathematical induction.
Problem 11.9
For each positive integer n let P (n) be the proposition 4n −1 is divisible by 3.
a. Write P (1). Is P (1) true?
b. Write P (k).
c. Write P (k + 1).
d. In a proof by mathematical induction that this divisibility property holds
for all integers n ≥ 1, what must be shown in the induction step?
Problem 11.10
For each positive integer n let P (n) be the proposition 23n − 1 is divisible by
7. Prove this property by mathematical induction.
Problem 11.11
Show that 2n < (n + 2)! for all integers n ≥ 0.
Problem 11.12
a. Use mathematical induction to show that n3 > 2n + 1 for all integers
n ≥ 2.
b. Use mathematical induction to show that n! > n2 for all integers n ≥ 4.
Problem 11.13
A sequence a1 , a2 , · · · is deﬁned by a1 = 3 and an = 7an−1 for n ≥ 2. Show
that an = 3 · 7n−1 for all integers n ≥ 1.
74.
74 FUNDAMENTALS OF MATHEMATICAL PROOFS
12 Project III: Elementary Number Theory
and Mathematical Proofs
Recall that the set of positive integers together with zero is denoted by IN.
The set of all integers is denoted by Z and the set of rational numbers is
denoted by Q.
I
We say that an integer n is even if and only if there exists an integer k
such that n = 2k. An integer n is said to be odd if and only if there exists
an integer k such that n = 2k + 1.
Problem 12.1
Let m and n be two integers.
a. Is 6m + 8n an even integer?
b. Is 6m + 4n2 + 3 odd?
Let a and b be two integers with a = 0. We say that b is divisible by a,
written a|b, if there exists an integer k such that b = ka. In this case we say
that a divides b, a is a factor of b, and b is a multiple of a. For example,
3 |7 whereas 3|12.
Problem 12.2
Prove the following theorem.
Theorem 12.1
Let a = 0, b = 0, and c be integers.
(i) If a|b and a|c then a|(b ± c).
(ii) If a|b then a|bc.
(iii) If a|b and b|c then a|c.
A positive integer p > 1 is called prime if 1 and p are the only divisors of p.
A number which is not prime is called a composite number. For example,
3 is prime whereas 10 is composite.
Problem 12.3
Let m and n be positive integers with m > n. Is m2 − n2 composite?
Problem 12.4
Write the ﬁrst 7 prime numbers.
75.
12 PROJECT III: ELEMENTARY NUMBER THEORY AND MATHEMATICAL PROOFS75
Problem 12.5
If a positive number p is composite then one can always write p as the product
of primes, where the prime factors are written in increasing order. This result
is known as the Fundamental Theorem of Arithmetic or the Unique
Factorization Theorem. Write the prime factorization of 180.
The following important theorem shows that if a number is not divisible by
any prime less than to its square root then the number must be prime.
Theorem 12.2
If n is a composite integer, then n has a prime divisor less than or equal to
√
n.
Proof.
Since n is composite, there is a divisor a of n such that 1 < a < n. Write
√ √ √ √
n = ab. If a > n √ b > n√
and then n = ab > n n = n, a false conclusion.
Thus, either a ≤ n or b ≤ n. Hence, n has a positive divisor which is
√
less than or equal to n. This divisor is either prime or, by the Fundamental
Theorem of Arithmetic has √ prime divisor. In either case, n has a prime
a
divisor less than or equal to n
Problem 12.6
Use the previous theorem to show that the number 101 is prime.
76.
76 FUNDAMENTALS OF MATHEMATICAL PROOFS
13 Project IV: The Euclidean Algorithm
Problem 13.1
Let a and b be two integers not both equal to zero. We say that d is the
greatest common divisor of a and b, written d = gcd(a, b), if d is the
largest integer such that d|a and d|b. If d = 1 then we say that a and b are
relatively prime. To ﬁnd d one writes the prime factorization of both a
and b, say a = pa1 pa2 · · · pan , b = pb1 pb2 · · · pbn , then
1 2 n 1 2 n
min(a1 ,b1 ) min(a2 ,b2 )
d = p1 p2 · · · pmin(an ,bn ) .
n
(i) Find gcd(120, 500).
(ii) Show that 17 and 22 are relatively prime.
Problem 13.2
We say that m is the least common multiple of two positive integers a
and b, written m = lcm(a, b), if m is the smallest positive integer that is
divisible by both a and b. Using the notation of the previous exercise m is
max(a1 ,b1 ) max(a2 ,b2 ) max(an ,bn )
given by m = p1 p2 · · · pn . Find lcm(120, 500).
Problem 13.3
Recall that a ≡ b mod n if and only if a − b = kn for some integer k.
(i) Show that if a ≡ b mod n and c ≡ d mod n then a + c ≡ b + d mod n.
(ii) Show that if a ≡ b mod n and c ≡ d mod n then ac ≡ bd mod n.
(iii) What are the solutions of the linear congruences 3x ≡ 4(mod7)?
Lemma 13.1 (Euclidean Algorithm)
Let a, b, q, and r be integers such that a = bq + r. Then gcd(a, b) = gcd(b, r).
Proof.
Let d1 = gcd(a, b) and d2 = gcd(b, r). We will show that d1 = d2 . Since d2 |b,
d2 |bq. Also d2 |r. Consequently d2 |(bq + r) that is d2 |a. Hence, d2 ≤ d1 . A
similar argument shows that d1 ≤ d2 . We conclude that d1 = d2
Using Lemma 13.1 we derive an algorithm, called the Euclidean Algo-
rithm, for ﬁnding the greatest common divisor of two non-negative integers
a and b with b = 0.
Dividing a by b we obtain
a = bq + r1 , where 0 ≤ r1 < b.
77.
13 PROJECT IV: THE EUCLIDEAN ALGORITHM 77
By Lemma 13.1 we have gcd(a, b) = gcd(b, r1 ). If r1 = 0 then we divide b by
r1 to obtain
b = r1 q1 + r2 , where 0 ≤ r2 < r1 .
Again by Lemma 13.1 we have gcd(b, r1 ) = gcd(r1 , r2 ). If r2 = 0 then we
divide r1 by r2 to obtain
r1 = r2 q2 + r3 , where 0 ≤ r3 < r2 .
By Lemma 13.1 we have gcd(r1 , r2 ) = gcd(r2 , r3 ). Repeating the above process,
ultimately, we will end up with rn = rn+1 qn+1 . In this case rn+1 = gcd(a, b).
Problem 13.4
a. Use the Euclidean algorithm to ﬁnd gcd(414, 662).
b. Use the Euclidean algorithm to ﬁnd gcd(287, 91).
78.
78 FUNDAMENTALS OF MATHEMATICAL PROOFS
14 Project V: Induction and the Algebra of
Matrices
In this section, we introduce the concept of a matrix. We also examine four
operations on matrices- equality, addition, scalar multiplication, and multi-
plication.
A matrix A of size m × n is a rectangular array of the form
a11 a12 ... a1n
a21 a22 ... a2n
A= ...
... ... ...
am1 am2 ... amn
where the aij ’s are the entries of the matrix, m is the number of rows, n
is the number of columns. The zero matrix 0 is the matrix whose entries
are all 0. The n × n identity matrix In is a square matrix whose main
diagonal consists of 1 s and the oﬀ diagonal entries are all 0. A matrix A can
be represented with the following compact notation A = (aij ). The ith row
of the matrix A is
[ai1 , ai2 , ..., ain ]
and the jth column is
a1j
a2j
.
.
.
amj
In what follows we discuss the basic arithmetic of matrices.
Two matrices are said to be equal if they have the same size and their cor-
responding entries are all equal. If the matrix A is not equal to the matrix
B we write A = B.
Problem 14.1
Find x1 , x2 and x3 such that
x1 + x2 + 2x3 0 1 9 0 1
2 3 2x1 + 4x2 − 3x3 = 2 3 1
4 3x1 + 6x2 − 5x3 5 4 0 5
79.
14 PROJECT V: INDUCTION AND THE ALGEBRA OF MATRICES 79
Problem 14.2
Solve the following matrix equation for a, b, c, and d
a−b b+c 8 1
=
3d + c 2a − 4d 7 6
Next, we introduce the operation of addition of two matrices. If A and B are
two matrices of the same size, then the sum A + B is the matrix obtained
by adding together the corresponding entries in the two matrices. Matrices
of diﬀerent sizes cannot be added.
Problem 14.3
Consider the matrices
2 1 2 1 2 1 0
A= ,B = ,C =
3 4 3 5 3 4 0
Compute, if possible, A + B, A + C and B + C.
If A is a matrix and c is a scalar, then the product cA is the matrix obtained
by multiplying each entry of A by c. Hence, −A = (−1)A. We deﬁne, A−B =
A + (−B). The matrix cIn is called a scalar matrix.
Problem 14.4
Consider the matrices
2 3 4 0 2 7
A= ,B =
1 2 1 1 −3 5
Compute A − 3B.
Problem 14.5
Let A be an m × n matrix. The transpose of A, denote by AT , is the n × m
whose columns are the rows of A. Find the transpose of the matrix
2 3 4
A= ,
1 2 1
80.
80 FUNDAMENTALS OF MATHEMATICAL PROOFS
Now, let A be a matrix of size m × n and entries aij ; B is a matrix of size
n × p and entries bij . Then the product matrix is a matrix of size m × p and
entries
cij = ai1 b1j + ai2 b2j + · · · + ain bnj
that is cij is obtained by multiplying componentwise the entries of the ith
row of A by the entries of the jth column of B. It is very important to keep
in mind that the number of columns of the ﬁrst matrix must be equal to the
number of rows of the second matrix; otherwise the product is undeﬁned.
Problem 14.6
Consider the matrices
4 1 4 3
1 2 4
A= , B = 0 −1 3 1
2 6 0
2 7 5 2
Compute, if possible, AB and BA.
Problem 14.7
Prove by induction on n ≥ 1 that
n
2 1 2n n2n−1
= .
0 2 0 2n
81.
Fundamentals of Set Theory
Set is the most basic term in mathematics and computer science. Hardly
any discussion in either subject can proceed without set or some synonym
such as class or collection. In this chapter we introduce the concept of sets
and its various operations and then study the properties of these operations.
15 Basic Deﬁnitions
We ﬁrst consider the following known as the barber puzzle:” The army
captain orders his company barber to shave all members of the company
provided they do not shave themselves. The barber is so busy at ﬁrst that
his own beard begins to be unsightly. Just as he lathers up, the impossibility
of his position strikes him: If he shaves himself, he disobeys the captain’s or-
der. If he does not shave himself, then by the captain’s order he is supposed
to shave himself.”
A situation like this is known as a paradox. To resolve the problem one has
to take the barber out of the company. Another well known paradox is
Russell’s Paradox. Deﬁne the set A = {X : X is a set, X ∈ X}.
Since A is a set then saying that A ∈ A will imply that A ∈ A by the deﬁni-
tion of A. Saying that A ∈ A means that A ∈ A by the deﬁnition of A. Thus
in either case the assumption that A is a set leads to an untenable paradox:
A ∈ A and A ∈ A. Hence, A is not a set.
Such a paradox indicated the necessity of a formal axiomatization of set the-
ory.
We deﬁne a set A as a collection of well-deﬁned objects (called elements or
members of A) such that for any given object x either one (but not both)
of the following holds:
81
82.
82 FUNDAMENTALS OF SET THEORY
• x belongs to A and we write x ∈ A.
• x does not belong to A, and in this case we write x ∈ A.
We denote sets by capital letters A, B, C, · · · and elements by lowercase let-
ters a, b, c, · · · Sets consisting of sets will be denoted by script letters.
There are two diﬀerent ways to represent a set. The ﬁrst one is to list,
without repetition, the elements of the set. The other way is to describe a
property that characterizes the elements of the set.
We deﬁne the empty set, denoted by ∅, to be the set with no elements.
Example 15.1
List the elements of the following sets.
a. {x|x is a real number such that x2 = 1}.
b. {x|x is an integer such that x2 − 3 = 0}.
Solution.
a. {−1, 1}.
b. ∅
Example 15.2
Use a property to give a description of each of the following sets.
a. {a, e, i, o, u}.
b. {1, 3, 5, 7, 9}.
Solution.
a. {x|x is a vowel}.
b. {n ∈ IN∗ |n is odd and less than 10}
Let A and B be two sets. We say that A is a subset of B, denoted by
A ⊆ B, if and only if every element of A is also an element of B. Symboli-
cally:
A ⊆ B ⇔ ∀x, x ∈ A implies x ∈ B
If there exists an element of A which is not in B then we write A ⊆ B.
Since the proposition x ∈ ∅ is always false then for any set A we have
∅ ⊆ A ⇔ ∀x, x ∈ ∅ implies x ∈ A
83.
15 BASIC DEFINITIONS 83
Example 15.3
Suppose that A = {2, 4, 6}, B = {2, 6}, and C = {4, 6}. Determine which of
these sets are subsets of which other of these sets.
Solution.
B ⊆ A and C ⊆ A
If sets A and B are represented as regions in the plane, relationships be-
tween A and B can be represented by pictures, called Venn diagram.
Example 15.4
Represent A ⊆ B using Venn diagram.
Solution.
Two sets A and B are said to be equal if and only if A ⊆ B and B ⊆ A.
We write A = B. Thus, to show that A = B it suﬃces to show the double
inclusions mentioned in the deﬁnition. For non-equal sets we write A = B.
Example 15.5
Determine whether each of the following pairs of sets are equal.
(a) {1, 3, 5} and {5, 3, 1}.
(b) {{1}} and {1, {1}}.
Solution.
(a) {1, 3, 5} = {5, 3, 1}.
(b) {{1}} = {1, {1}} since 1 ∈ {{1}}
84.
84 FUNDAMENTALS OF SET THEORY
Let A and B be two sets. We say that A is a proper subset of B, de-
noted by A ⊂ B, if A ⊆ B and A = B. Thus, to show that A is a proper
subset of B we must show that every element of A is an element of B and
there is an element of B which is not in A.
Example 15.6
Order the sets of numbers: Z , IR, Q, IN using ⊂
I
Solution.
IN ⊂ Z ⊂ Q ⊂ IR
I
Example 15.7
Determine whether each of the following statements is true or false.
(a) x ∈ {x} (b) {x} ⊆ {x} (c) {x} ∈ {x}
(d) {x} ∈ {{x}} (e) ∅ ⊆ {x} (f) ∅ ∈ {x}
Solution.
(a) True (b) True (c) False (d) True (e) True (f) False
If U is a given set whose subsets are under consideration, then we call U
a universal set.
Let U be a universal set and A, B be two subsets of U. The absolute com-
plement of A is the set
Ac = {x ∈ U |x ∈ A}.
The relative complement of A with respect to B is the set
B − A = {x ∈ U |x ∈ B and x ∈ A}.
Example 15.8
Let U = IR. Consider the sets A = {x ∈ IR|x < −1 or x > 1} and
B = {x ∈ IR|x ≤ 0}. Find
a. Ac .
b. B − A.
Solution.
a. Ac = [−1, 1].
85.
15 BASIC DEFINITIONS 85
b. B − A = [−1, 0]
Let A and B be two sets. The union of A and B is the set
A ∪ B = {x|x ∈ A or x ∈ B}.
where the ’or’ is inclusive. This defenition can be extended to more than two
sets. More precisely, if A1 , A2 , · · · , are sets then
∪∞ An = {x|x ∈ Ai f or some i}.
n=1
Let A and B be two sets. The intersection of A and B is the set
A ∩ B = {x|x ∈ A and x ∈ B}.
If A∩B = ∅ we say that A and B are disjoint sets. Given the sets A1 , A2 , · · · ,
we deﬁne
∩∞ An = {x|x ∈ Ai f or all i}.
n=1
Example 15.9
Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}.
a. Find A ∪ (B ∩ C), (A ∪ B) ∩ C, and (A ∪ B) ∩ (A ∪ C). Which of
these sets are equal?
b. Find A ∩ (B ∪ C), (A ∩ B) ∪ C, and (A ∩ B) ∪ (A ∩ C). Which of these
sets are equal?
c. Find A − (B − C) and (A − B) − C. Are these sets equal?
Solution.
a. A ∪ (B ∩ C) = A, (A ∪ B) ∩ C = {b, c}, (A ∪ B) ∩ (A ∪ C) = {b, c} =
(A ∪ B) ∩ C.
b. A ∩ (B ∪ C) = {b, c}, (A ∩ B) ∪ C = C, (A ∩ B) ∪ (A ∩ C) = {b, c} =
(A ∩ B) ∪ C.
c. A − (B − C) = A and (A − B) − C = {a} = A − (B − C).
Example 15.10
1
For each n ≥ 1, let An = {x ∈ IR : x < 1 + n }. Show that
∩∞ An = {x ∈ IR : x ≤ 1}.
n=1
86.
86 FUNDAMENTALS OF SET THEORY
Solution.
The proof is by double inclusions method. Let y ∈ {x ∈ IR : x ≤ 1}. Then
for all positive integer n we have y ≤ 1 < 1 + n . That is, y ∈ ∩∞ An . This
1
n=1
shows that {x ∈ IR : x ≤ 1} ⊆ ∩∞ An .
n=1
Conversely, let y ∈ ∩∞ An . Then y < 1 + n for all n ≥ 1. Now take the limit
n=1
1
of both sides as n → ∞ to obtain y ≤ 1. That is, y ∈ {x ∈ IR : x ≤ 1}. This
shows that ∩∞ An ⊆ {x ∈ IR : x ≤ 1}.
n=1
Example 15.11
The symmetric diﬀerence of A and B, denoted by A∆B, is the set contain-
ing those elements in either A or B but not both. Find A∆B if A = {1, 3, 5}
and B = {1, 2, 3}.
Solution.
A∆B = {2, 5}
The notation (a1 , a2 , · · · , an ) is called an ordered n-tuples. We say that
two n-tuples (a1 , a2 , · · · , an ) and (b1 , b2 , · · · , bn ) are equal if and only if a1 =
b1 , a2 = b2 , · · · , an = bn .
Given n sets A1 , A2 , · · · , An the Cartesian product of these sets is the set
A1 × A2 × · · · × An = {(a1 , a2 , · · · , an ) : a1 ∈ A1 , a2 ∈ A2 , · · · , an ∈ An }
Example 15.12
Let A = {x, y}, B = {1, 2, 3}, and C = {a, b}. Find
a. A × B × C.
b. (A × B) × C.
Solution.
a.
A × B × C = {(x, 1, a), (x, 2, a), (x, 3, a), (y, 1, a), (y, 2, a),
(y, 3, a), (x, 1, b), (x, 2, b), (x, 3, b), (y, 1, b)
(y, 2, b), (y, 3, b)}
b.
(A × B) × C = {((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a),
((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b)
((y, 2), b), ((y, 3), b)}
87.
15 BASIC DEFINITIONS 87
Next, we introduce one more special kind of sets, denoted by Σ∗ . An al-
phabet is a ﬁnite nonempty set Σ whose members are called letters and
with the restrictions that Σ does not contain letters which are themselves
strings beginning with other letters of Σ. Thus, Σ = {a, b, c, ca} is not an
alphabet. A word is any ﬁnite string of letters from Σ. We denote the set of
all words using letters from Σ by Σ∗ . Any subset of Σ∗ is called a language.
For example, if Σ consists of the twenty six letters of the english alphabet,
then the American language can be deﬁned as the subset of Σ∗ consisting of
words in the latest edition of the Webster’s World dictionary of the American
Language.
The empty word or the null word is the string with no letters. It is de-
noted by .
We deﬁne the length of a word w to be the number of letters from Σ in
w and we write |w|. Note that in order to deﬁne the length of a word the
restriction given in the deﬁnition is needed. To be more precise, suppose that
Σ = {a, b, ab}. Then what is the length of the word aab? Is this a word with
two letters a and ab or three letters a, a, and b? So obviously there is no way
to tell. This ambiguity is resolved by making the restriction stated in the
deﬁnition above.
Finally, by Σn we mean the set of all words over Σ of length n. That is, Σn
is the cartesian product of n copies of Σ.
Example 15.13
Let Σ = {a, b}. List all the elements of the set
A = {w ∈ Σ∗ : |w| = 2}.
Solution.
A = {aa, ab, ba, bb}
88.
88 FUNDAMENTALS OF SET THEORY
Review Problems
Problem 15.1
Which of the following sets are equal?
a. {a, b, c, d}
b. {d, e, a, c}
c. {d, b, a, c}
d. {a, a, d, e, c, e}
Problem 15.2
Let A = {c, d, f, g}, B = {f, j}, and C = {d, g}. Answer each of the following
questions. Give reasons for your answers.
a. Is B ⊆ A?
b. Is C ⊆ A?
c. Is C ⊆ C?
d. Is C is a proper subset of A?
Problem 15.3
a. Is 3 ∈ {1, 2, 3}?
b. Is 1 ⊆ {1}?
c. Is {2} ∈ {1, 2}?
d. Is {3} ∈ {1, {2}, {3}}?
e. Is 1 ∈ {1}?
f. Is {2} ⊆ {1, {2}, {3}}?
g. Is {1} ⊆ {1, 2}?
h. Is 1 ∈ {{1}, 2}?
i. Is {1} ⊆ {1, {2}}?
j. Is {1} ⊆ {1}?
Problem 15.4
Let A = {b, c, d, f, g} and B = {a, b, c}. Find each of the following:
a. A ∪ B.
b. A ∩ B.
c. A − B.
d. B − A.
Problem 15.5
Indicate which of the following relationships are true and which are false:
a. Z + ⊆ Q.
I
89.
15 BASIC DEFINITIONS 89
b. IR− ⊂ Q.
I
c. Q ⊂ Z .
I
d. Z + ∪ Z − = Z .
e. Q ∩ IR = Q.
I I
f. Q ∪ Z = Z .
I
g. Z + ∩ IR = Z +
h. Z ∪ Q = Q.
I I
Problem 15.6
Let A = {x, y, z, w} and B = {a, b}. List the elements of each of the following
sets:
a. A × B
b. B × A
c. A × A
d. B × B.
Problem 15.7
Let Σ = {x, y} be an alphabet.
a. Let L1 be the language consisting of all strings over Σ that are palindromes
and have length ≤ 4. List the elements L1 .
b. Let L2 be the language consisting of all strings over Σ that begins with x
and have length ≤ 3. List the elements L2 .
c. Let L3 be the language consisting of all strings over Σ with length ≤ 3
and for which all the x s appear to the left of all the y s. List the elements
L3 .
d. List the elements of Σ4 , the set of all strings of length 4 over Σ.
e. Let A = Σ3 ∪ Σ4 . Describe A, B, and A ∪ B in words.
90.
90 FUNDAMENTALS OF SET THEORY
16 Properties of Sets
The following exercise shows that the operation ⊆ is reﬂexive and transitive,
concepts that will be discussed in the next chapter.
Example 16.1
a. Suppose that A, B, C are sets such that A ⊆ B and B ⊆ C. Show that
A ⊆ C.
b. Find two sets A and B such that A ∈ B and A ⊆ B.
c. Show that A ⊆ A.
Solution.
a. We need to show that every element of A is an element of C. Let x ∈ A.
Since A ⊆ B then x ∈ B. But B ⊆ C so that x ∈ C.
b. A = {x} and B = {x, {x}}.
c. The proposition if x ∈ A then x ∈ A is always true. Thus, A ⊆ A
Theorem 16.1
Let A and B be two sets. Then
a. A ∩ B ⊆ A and A ∩ B ⊆ B.
b. A ⊆ A ∪ B and B ⊆ A ∪ B.
Proof.
a. If x ∈ A ∩ B then x ∈ A and x ∈ B. This still imply that x ∈ A. Hence,
A ∩ B ⊆ A. A similar argument holds for A ∩ B ⊆ B.
b. The proposition ”if x ∈ A then x ∈ A ∪ B” is always true. Hence,
A ⊆ A ∪ B. A similar argument holds for B ⊆ A ∪ B
Theorem 16.2
Let A be a subset of a universal set U. Then
a. ∅c = U.
b. U c = ∅.
c. (Ac )c = A.
d. A ∪ Ac = U.
e. A ∩ Ac = ∅.
Proof.
a. If x ∈ U then x ∈ U and x ∈ ∅. Thus, U ⊆ ∅c . Conversely, suppose that
x ∈ ∅c . Then x ∈ U and x ∈ ∅. This implies that x ∈ U. Hence, ∅c ⊆ U.
91.
16 PROPERTIES OF SETS 91
b. It is always true that ∅ ⊆ U c . Conversely, the proposition ”x ∈ U and
x ∈ U implies x ∈ ∅” is vacuously true since the hypothesis is false. This
says that U c ⊆ ∅.
c. Let x ∈ (Ac )c . Then x ∈ U and x ∈ Ac . That is, x ∈ U and (x ∈ U or
x ∈ A). Since x ∈ U then x ∈ A. Hence (Ac )c ⊆ A. Conversely, suppose that
x ∈ A. Then x ∈ U and x ∈ A. That is, x ∈ U and x ∈ Ac . Thus, x ∈ (Ac )c .
This shows that A ⊆ (Ac )c .
d. It is clear that A ∪ Ac ⊆ U. Conversely, suppose that x ∈ U. Then either
x ∈ A or x ∈ A. But this is the same as saying that x ∈ A ∪ Ac .
e. By deﬁnition ∅ ⊆ A ∩ Ac . Conversely, the conditional proposition ”x ∈ A
and x ∈ A implies x ∈ ∅” is vacuously true since the hypothesis is false. This
shows that A ∩ Ac ⊆ ∅
Theorem 16.3
If A and B are subsets of U then
a. A ∪ U = U.
b. A ∪ A = A.
c. A ∪ ∅ = A.
d. A ∪ B = B ∪ A.
e. (A ∪ B) ∪ C = A ∪ (B ∪ C).
Proof.
a. Clearly, A ∪ U ⊆ U. Conversely, let x ∈ U . Then deﬁnitely, x ∈ A ∪ U.
That is, U ⊆ A ∪ U.
b. If x ∈ A then x ∈ A or x ∈ A. That is, x ∈ A ∪ A and consequently
A ⊆ A ∪ A. Conversely, if x ∈ A ∪ A then x ∈ A. Hence, A ∪ A ⊆ A.
c. If x ∈ A ∪ ∅ then x ∈ A since x ∈ ∅. Thus, A ∪ ∅ ⊆ A. Conversely, if x ∈ A
then x ∈ A or x ∈ ∅. Hence, A ⊆ A ∪ ∅.
d. If x ∈ A ∪ B then x ∈ A or x ∈ B. But this is the same thing as saying
x ∈ B or x ∈ A. That is, x ∈ B ∪ A. Now interchange the roles of A and B
to show that B ∪ A ⊆ A ∪ B.
e. Let x ∈ (A ∪ B) ∪ C. Then x ∈ (A ∪ B) or x ∈ C. Thus, (x ∈ A or x ∈ B)
or x ∈ C. This implies x ∈ A or (x ∈ B or x ∈ C). Hence, x ∈ A ∪ (B ∪ C).
The converse is similar
Theorem 16.4
Let A and B be subsets of U . Then
a. A ∩ U = A.
92.
92 FUNDAMENTALS OF SET THEORY
b. A ∩ A = A.
c. A ∩ ∅ = ∅.
d. A ∩ B = B ∩ A.
e. (A ∩ B) ∩ C = A ∩ (B ∩ C).
Proof.
a. If x ∈ A ∩ U then x ∈ A. That is , A ∩ U ⊆ A. Conversely, let x ∈ A.
Then deﬁnitely, x ∈ A and x ∈ U. That is, x ∈ A ∩ U. Hence, A ⊆ A ∩ U.
b. If x ∈ A then x ∈ A and x ∈ A. That is, A ⊆ A ∩ A. Conversely, if
x ∈ A ∩ A then x ∈ A. Hence, A ∩ A ⊆ A.
c. Clearly ∅ ⊆ A ∩ ∅. Conversely, if x ∈ A ∩ ∅ then x ∈ ∅. Hence, A ∩ ∅ ⊆ ∅.
d. If x ∈ A ∩ B then x ∈ A and x ∈ B. But this is the same thing as saying
x ∈ B and x ∈ A. That is, x ∈ B ∩ A. Now interchange the roles of A and B
to show that B ∩ A ⊆ A ∩ B.
e. Let x ∈ (A ∩ B) ∩ C. Then x ∈ (A ∩ B) and x ∈ C. Thus, (x ∈ A and
x ∈ B) and x ∈ C. This implies x ∈ A and (x ∈ B and x ∈ C). Hence,
x ∈ A ∩ (B ∩ C). The converse is similar
Theorem 16.5
If A, B, and C are subsets of U then
a. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
b. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Proof.
a. Let x ∈ A ∩ (B ∪ C). Then x ∈ A and x ∈ B ∪ C. Thus, x ∈ A and (x ∈ B
or x ∈ C). This implies that (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C).
Hence, x ∈ A ∩ B or x ∈ A ∩ C, i.e. x ∈ (A ∩ B) ∪ (A ∩ C). The converse is
similar.
b. Let x ∈ A ∪ (B ∩ C). Then x ∈ A or x ∈ B ∩ C. Thus, x ∈ A or (x ∈ B
and x ∈ C). This implies that (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C).
Hence, x ∈ A ∪ B and x ∈ A ∪ C, i.e. x ∈ (A ∪ B) ∩ (A ∪ C). The converse
is similar
Theorem 16.6 (De Morgan’s Laws)
Let A and B be subsets of U then
a. (A ∪ B)c = Ac ∩ B c .
b. (A ∩ B)c = Ac ∪ B c .
93.
16 PROPERTIES OF SETS 93
Proof.
a. Let x ∈ (A ∪ B)c . Then x ∈ U and x ∈ A ∪ B. Hence, x ∈ U and (x ∈ A
and x ∈ B). This implies that (x ∈ U and x ∈ A) and (x ∈ U and x ∈ B).
It follows that x ∈ Ac ∩ B c . Now, go backward for the converse.
b. Let x ∈ (A ∩ B)c . Then x ∈ U and x ∈ A ∩ B. Hence, x ∈ U and (x ∈ A
or x ∈ B). This implies that (x ∈ U and x ∈ B) or (x ∈ U and x ∈ A). It
follows that x ∈ Ac ∪ B c . The converse is similar
Theorem 16.7
Suppose that A ⊆ B. Then
a. A ∩ B = A.
b. A ∪ B = B.
Proof.
a. If x ∈ A ∩ B then by the deﬁnition of intersection of two sets we have
x ∈ A. Hence, A ∩ B ⊆ A. Conversely, if x ∈ A then x ∈ B as well since
A ⊆ B. Hence, x ∈ A ∩ B. This shows that A ⊆ A ∩ B.
b. If x ∈ A ∪ B then x ∈ A or x ∈ B. Since A ⊆ B then x ∈ B. Hence,
A∪B ⊆ B. Conversely, if x ∈ B then x ∈ A∪B. This shows that B ⊆ A∪B.
Example 16.2
Let A and B be arbitrary sets. Show that (A − B) ∩ B = ∅.
Solution.
Suppose not. That is, suppose (A − B) ∩ B = ∅. Then there is an element x
that belongs to both A − B and B. By the deﬁnition of A − B we have that
x ∈ B. Thus, x ∈ B and x ∈ B which is a contradiction
A collection of nonempty subsets {A1 , A2 , · · · , An } of A is said to be a par-
tition of A if and only if
(i) A = ∪n Ak .
k=1
(ii) Ai ∩ Aj = ∅ for all i = j.
Example 16.3
Let A = {1, 2, 3, 4, 5, 6}, A1 = {1, 2}, A2 = {3, 4}, A3 = {5, 6}. Show that
{A1 , A2 , A3 } is a partition of A.
94.
94 FUNDAMENTALS OF SET THEORY
Solution.
(i) A1 ∪ A2 ∪ A3 = A.
(ii) A1 ∩ A2 = A1 ∩ A3 = A2 ∩ A3 = ∅.
The number of elements of a set is called the cardinality of the set. We
write |A| to denote the cardinality of the set A. If A has a ﬁnite cardinality
we say that A is a ﬁnite set. Otherwise, it is called inﬁnite.
Example 16.4
What is the cardinality of each of the following sets.
(a) ∅.
(b) {∅}.
(c) {a, {a}, {a, {a}}}.
Solution.
(a) |∅| = 0
(b) |{∅}| = 1
(c) |{a, {a}, {a, {a}}}| = 3
Let A be a set. The power set of A, denoted by P(A), is the empty set
together with all possible subsets of A.
Example 16.5
Find the power set of A = {a, b, c}.
Solution.
P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c},
{b, c}, {a, b, c}}
Theorem 16.8
If A ⊆ B then P(A) ⊆ P(B).
Proof.
Let X ∈ P(A). Then X ⊆ A. Since A ⊆ B then X ⊆ B. Hence, X ∈ P(B)
Example 16.6
a. Use induction to show that if |A| = n then |P(A)| = 2n .
b. If P(A) has 256 elements, how many elements are there in A?
95.
16 PROPERTIES OF SETS 95
Solution.
a. If n = 0 then A = ∅ and in this case P(A) = {∅}. Thus |P(A)| = 1.
As induction hypothesis, suppose that if |A| = n then |P(A)| = 2n . Let
B = A ∪ {an+1 }. Then P(B) consists of all subsets of A and all subsets of A
with the element an+1 added to them. Hence, |P(B)| = 2n +2n = 2·2n = 2n+1 .
b. Since |P(A)| = 256 = 28 then |A| = 8
96.
96 FUNDAMENTALS OF SET THEORY
Review Problems
Problem 16.1
Let A, B, and C be sets. Prove that if A ⊆ B then A ∩ C ⊆ B ∩ C.
Problem 16.2
Find sets A, B, and C such that A ∩ C = B ∩ C but A = B.
Problem 16.3
Find sets A, B, and C such that A ∩ C ⊆ B ∩ C and A ∪ C ⊆ B ∪ C but
A = B.
Problem 16.4
Let A and B be two sets. Prove that if A ⊆ B then B c ⊆ Ac .
Problem 16.5
Let A, B, and C be sets. Prove that if A ⊆ C and B ⊆ C then A ∪ B ⊆ C.
Problem 16.6
Let A, B, and C be sets. Show that A × (B ∪ C) = (A × B) ∪ (A × C).
Problem 16.7
Let A, B, and C be sets. Show that A × (B ∩ C) = (A × B) ∩ (A × C).
Problem 16.8
a. Is the number 0 in ∅? Why?
b. Is ∅ = {∅}? Why?
c. Is ∅ ∈ {∅}? Why?
Problem 16.9
Let A and B be two sets. Prove that (A − B) ∩ (A ∩ B) = ∅.
Problem 16.10
Let A and B be two sets. Show that if A ⊆ B then A ∩ B c = ∅.
Problem 16.11
Let A, B and C be three sets. Prove that if A ⊆ B and B ∩ C = ∅ then
A ∩ C = ∅.
97.
16 PROPERTIES OF SETS 97
Problem 16.12
Find two sets A and B such that A ∩ B = ∅ but A × B = ∅.
Problem 16.13
Suppose that A = {1, 2} and B = {2, 3}. Find each of the following:
a. P(A ∩ B).
b. P(A).
c. P(A ∪ B).
d. P(A × B).
Problem 16.14
a. Find P(∅).
b. Find P(P(∅)).
c. Find P(P(P(∅))).
Problem 16.15
Determine which of the following statements are true and which are false.
Prove each statement that is true and give a counterexample for each state-
ment that is false.
a. P(A ∪ B) = P(A) ∪ P(B).
b. P(A ∩ B) = P(A) ∩ P(B).
c. P(A) ∪ P(B) ⊆ P(A ∪ B).
d. P(A × B) = P(A) × P(B).
98.
98 FUNDAMENTALS OF SET THEORY
17 Project VI: Boolean Algebra
A Boolean algebra is a nonempty set S together with two operations ⊕
and that satisfy the following axioms:
• a ⊕ b ∈ S and a b ∈ S for all a, b ∈ S.
• a ⊕ b = b ⊕ a and a b = b a, ∀a, b ∈ S.
• a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c and a (b c) = (a b) c), ∀a, b, c ∈ S.
• a ⊕ (b c) = (a ⊕ b) (a ⊕ c) and a (b ⊕ c) = a b ⊕ a c ∀a, b, c ∈ S.
• There exist distinct elements 0 and 1 in S such that a⊕0 = a and a 1 = a
∀a ∈ S.
• For each a ∈ S there exits an element a such that a ⊕ a = 1 and a a = 0.
We call a the complement or the negation of a.
We write (S, ⊕, ).
Problem 17.1
Show that if S is a collection of propositions with ﬁnite propositional variables
then (S, ∨, ∧) is a Boolean algebra.
Problem 17.2
Show that for a given nonempty set S, (P(S), ∪, ∩) is a Boolean algebra.
99.
Relations and Functions
The reader is familiar with many relations which are used in mathematics
and computer science, i.e. ”is a subset of”, ” is less than” and so on.
One frequently wants to compare or contrast various members of a set, per-
haps to arrange them in some appropriate order or to group together those
with similar properties. The mathematical framework to describe this kind
of organization of sets is the theory of relations.
There are three kinds of relations which we discuss in this chapter: (i) equiv-
alence relations, (ii) order relations, (iii) functions.
18 Equivalence Relations
Let A be a given set. An ordered pair (a, b) of elements in A is deﬁned
to be the set {a, {a, b}}. The element a (resp. b) is called the ﬁrst (resp.
second) component.
Example 18.1
a. Show that if a = b then (a, b) = (b, a).
b. Show that (a, b) = (c, d) if and only if a = c and b = d.
Solution.
a. If a = b then {a, {a, b}} = {b, {a, b}}. That is, (a, b) = (b, a).
b. (a, b) = (c, d) if and only if {a, {a, b} = {c, {c, d}} and this is equivalent
to a = c and {a, b} = {c, d} by the deﬁnition of equality of sets. Thus, a = c
and b = d.
Example 18.2
Find x and y such that (x + y, 0) = (1, x − y).
99
100.
100 RELATIONS AND FUNCTIONS
Solution.
By the previous exercise we have the system
x + y = 1
x − y = 0
1 1
Solving by the method of elimination one ﬁnds x = 2
and y = 2 .
If A and B are sets, we let A × B denote the set of all ordered pairs (a, b)
where a ∈ A and b ∈ B. We call A × B the Cartesian product of A and
B.
Example 18.3
a. Show that if A is a set with m elements and B is a set of n elements then
A × B is a set of mn elements.
b. Show that if A × B = ∅ then A = ∅ or B = ∅.
Solution.
a. Consider an ordered pair (a, b). There are m possibilities for a. For each
ﬁxed a, there are n possibilities for b. Thus, there are m × n ordered pairs
(a, b). That is, |A × B| = mn.
b. We use the proof by contrapositive. Suppose that A = ∅ and B = ∅. Then
there is at least an a ∈ A and an element b ∈ B. That is, (a, b) ∈ A × B
and this shows that A × B = ∅. A contradiction to the assumption that
A×B =∅
Example 18.4
Let A = {1, 2}, B = {1}. Show that A × B = B × A.
Solution.
We have A × B = {(1, 1), (2, 1)} = {(1, 1), (1, 2)} = B × A.
A binary relation R from a set A to a set B is a subset of A × B. If
(a, b) ∈ R we write aRb and we say that a is related to b. If a is not related
to B we write a Rb. In case A = B we call R a binary relation on A.
The set
Dom(R) = {a ∈ A|(a, b) ∈ R f or some b ∈ B}
is called the domain of R. The set
Range(R) = {b ∈ B|(a, b) ∈ R f or some a ∈ A}
is called the range of R.
101.
18 EQUIVALENCE RELATIONS 101
Example 18.5
a. Let A = {2, 3, 4} and B = {3, 4, 5, 6, 7}. Deﬁne the relation R by aRb if
and only if a divides b. Find, R, Dom(R), Range(R).
b. Let A = {1, 2, 3, 4}. Deﬁne the relation R by aRb if and only if a ≤ b.
Find, R, Dom(R), Range(R).
Solution.
a. R = {(2, 4), (2, 6), (3, 3), (3, 6), (4, 4)}, Dom(R) = {2, 3, 4}, and Range(R) =
{3, 4, 6}.
b. R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}, Dom(R) =
A, Range(R) = A.
A function is a special case of a relation. A function from A to B, de-
noted by f : A → B, is a relation from A to B such that for every x ∈ A
there is a unique y ∈ B such that (x, y) ∈ f. The element y is called the
image of x and we write y = f (x). The set A is called the domain of f and
the set of all images of f is called the range of f. Functions will be discussed
in more details in Section 4.3.
Example 18.6
a. Show that the relation
f = {(1, a), (2, b), (3, a)}
deﬁnes a function from A = {1, 2, 3} to B = {a, b, c}. Find its range.
b. Show that the relation f = {(1, a), (2, b), (3, c), (1, b)} does not deﬁne a
function from A = {1, 2, 3} to B = {a, b, c}.
Solution.
a. Note that each element of A has exactly one image. Hence, f is a function
with domain A and range Range(f ) = {a, b}.
b. The relation f does not deﬁne a function since the element 1 has two
images, namely a and b.
An informative way to picture a relation on a set is to draw its digraph. To
draw a digraph of a relation on a set A, we ﬁrst draw dots or vertices to
represent the elements of A. Next, if (a, b) ∈ R we draw an arrow (called a
directed edge) from a to b. Finally, if (a, a) ∈ R then the directed edge is
simply a loop.
102.
102 RELATIONS AND FUNCTIONS
Example 18.7
Draw the directed graph of the relation in part (b) of Problem 18.5.
Solution.
Next we discuss three ways of building new relations from given ones. Let R
be a relation from a set A to a set B. The inverse of R is the relation R−1
from Range(R) to Dom(R) such that
R−1 = {(b, a) ∈ B × A : (a, b) ∈ R}.
Example 18.8
Let R = {(1, y), (1, z), (3, y)} be a relation from A = {1, 2, 3} to B =
{x, y, z}.
a. Find R−1 .
b. Compare (R−1 )−1 and R.
Solution.
a. R−1 = {(y, 1), (z, 1), (y, 3)}.
b. (R−1 )−1 = R.
Let R and S be two relations from a set A to a set B. Then we deﬁne
the relations R ∪ S and R ∩ S by
R ∪ S = {(a, b) ∈ A × B|(a, b) ∈ R or (a, b) ∈ S},
and
R ∩ S = {(a, b) ∈ A × B|(a, b) ∈ A and (a, b) ∈ B}.
103.
18 EQUIVALENCE RELATIONS 103
Example 18.9
Given the following two relations from A = {1, 2, 4} to B = {2, 6, 8, 10} :
aRb if and only if a|b.
aSb if and only if b − 4 = a.
List the elements of R, S, R ∪ S, and R ∩ S.
Solution.
R = {(1, 2), (1, 6), (1, 8), (1, 10), (2, 2), (2, 6), (2, 8), (2, 10), (4, 8)}
S = {(2, 6), (4, 8)}
R∪S = R
R∩S = S
Now, If we have a relation R from A to B and a relation S from B to C
we can deﬁne the relation S ◦ R, called the composition relation, to be the
relation from A to C deﬁned by
S ◦ R = {(a, c)|(a, b) ∈ R and (b, c) ∈ S f or some b ∈ B}.
Example 18.10
Let
R = {(1, 2), (1, 6), (2, 4), (3, 4), (3, 6), (3, 8)}
S = {(2, u), (4, s), (4, t), (6, t), (8, u)}
Find S ◦ R.
Solution.
S ◦ R = {(1, u), (1, t), (2, s), (2, t), (3, s), (3, t), (3, u)}
We next deﬁne four types of binary relations. A relation R on a set A is
called reﬂexive if (a, a) ∈ R for all a ∈ A. In this case, the digraph of R has
a loop at each vertex.
Example 18.11
a. Show that the relation a ≤ b on the set A = {1, 2, 3, 4} is reﬂexive.
b. Show that the relation on IR deﬁned by aRb if and only if a < b is not
reﬂexive.
104.
104 RELATIONS AND FUNCTIONS
Solution.
a. By Problem 242, each vertex has a loop.
b. Indeed, for any real number a we have a − a = 0 and not a − a < 0.
A relation R on A is called symmetric if whenever (a, b) ∈ R then we
must have (b, a) ∈ R. The digraph of a symmetric relation has the property
that whenever there is a directed edge from a to b, there is also a directed
edge from b to a.
Example 18.12
a. Let A = {a, b, c, d} and R = {(a, a), (b, c), (c, b), (d, d)}. Show that R is
symmetric.
b. Let IR be the set of real numbers and R be the relation aRb if and only
if a < b. Show that R is not symmetric.
Solution.
a. bRc and cRb so R is symmetric.
b. 2 < 4 but 4 < 2.
A relation R on a set A is called antisymmetric if whenever (a, b) ∈ R
and a = b then (b, a) ∈ R. The digraph of an antisymmetric relation has the
property that between any two vertices there is at most one directed edge.
Example 18.13
a. Let IN be the set of nonnegative integers and R the relation aRb if and
only if a divides b. Show that R is antisymmetric.
b. Let A = {a, b, c, d} and R = {(a, a), (b, c), (c, b), (d, d)}. Show that R is
not antisymmetric.
Solution.
a. Suppose that a|b and b|a. We must show that a = b. Indeed, by the def-
inition of division, there exist positive integers k1 and k2 such that b = k1 a
and a = k2 b. This implis that a = k2 k1 a and hence k1 k2 = 1. Since k1 and k2
are positive integers then we must have k1 = k2 = 1. Hence, a = b.
b. bRc and cRb with b = c.
A relation R on a set A is called transitive if whenever (a, b) ∈ R and
(b, c) ∈ R then (a, c) ∈ R. The digraph of a transitive relation has the prop-
erty that whenever there are directed edges from a to b and from b to c then
there is also a directed edge from a to c.
105.
18 EQUIVALENCE RELATIONS 105
Example 18.14
a. Let A = {a, b, c, d} and R = {(a, a), (b, c), (c, b), (d, d)}. Show that R is
not transitive.
b. Let Z be the set of integers and R the relation aRb if a divides b. Show
that R is transitive.
Solution.
a. (b, c) ∈ R and (c, b) ∈ R but (b, b) ∈ R.
b. Suppose that a|b and b|c. Then there exist integers k1 and k2 such that
b = k1 a and c = k2 b. Thus, c = (k1 k2 )a which means that a|c.
Now, let A1 , A2 , · · · , An be a partition of a set A. That is, the Ai s are
subsets of A that satisfy
(i) ∪n Ai = A
i=1
(ii) Ai ∩ Aj = ∅ for i = j.
Deﬁne on A the binary relation x R y if and only if x and y belongs to the
same set Ai for some 1 ≤ i ≤ n.
Theorem 18.1
The relation R deﬁned above is reﬂexive, symmetric, and transitive.
Proof.
• R is reﬂexive: If x ∈ A then by (i) x ∈ Ak for some 1 ≤ k ≤ n. Thus, x
and x belong to Ak so that x R x.
• R is symmetric: Let x, y ∈ A such that x R y. Then there is an index k
such that x, y ∈ Ak . But then y, x ∈ Ak . That is, y R x.
• R is transitive: Let x, y, z ∈ A such that x R y and y R z. Then there exist
indices i and j such that x, y ∈ Ai and y, z ∈ Aj . Since y ∈ Ai ∩ Aj then
by (ii) we must have i = j. This implies that x, y, z ∈ Ai and in particular
x, z ∈ Ai . Hence, x R z.
A relation that is reﬂexive, symmetric, and transitive on a set A is called
an equivalence relation on A. For example, the relation ”=” is an equiv-
alence relation on IR.
Example 18.15
Let Z be the set of integers and n ∈ Z . Let R be the relation on Z deﬁned
by aRb if a − b is a multiple of n. We denote this relation by a ≡ b (mod n)
read ”a congruent to b modulo n.” Show that R is an equivalence relation
on Z .
106.
106 RELATIONS AND FUNCTIONS
Solution.
≡ is reﬂexive: For all a ∈ Z , a − a = 0 · n. That is, a ≡ a (mod n).
≡ is symmetric: Let a, b ∈ Z such that a ≡ b (mod n). Then there is an
integer k such that a − b = kn. Multiply both sides of this equality by (−1)
and letting k = −k we ﬁnd that b − a = k n. That is b ≡ a (mod n).
≡ is transitive: Let a, b, c ∈ Z be such that a ≡ b (mod n) and b ≡ c (mod n).
Then there exist integers k1 and k2 such that a − b = k1 n and b − c = k2 n.
Adding these equalities together we ﬁnd a − c = kn where k = k1 + k2 ∈ Z
which shows that a ≡ c (mod n).
Theorem 18.2
Let R be an equivalence relation on A. For each a ∈ A let
[a] = {x ∈ A|xRa}
A/R = {[a]|a ∈ A}.
Then the union of all the elements of A/R is equal to A and the intersection
of any two distinct members of A/R is the empty set. That is, the family
A/R forms a partition of A.
Proof.
By the deﬁnition of [a] we have that [a] ⊆ A. Hence, ∪a∈A [a] ⊆ A. We
next show that A ⊆ ∪a∈A [a]. Indeed, let a ∈ A. Since A is reﬂexive then
a ∈ [a] and consequently a ∈ ∪b∈A [b]. Hence, A ⊆ ∪b∈A [b]. It follows that
A = ∪a∈A [a]. This establishes (i).
It remains to show that if [a] = [b] then [a] ∩ [b] = ∅ for a, b ∈ A. Suppose the
contrary. That is, suppose [a] ∩ [b] = ∅. Then there is an element c ∈ [a] ∩ [b].
This means that c ∈ [a] and c ∈ [b]. Hence, a R c and b R c. Since R is
symmetric and transitive then a R b. We will show that the conclusion a R b
leads to [a] = [b]. The proof is by double inclusions. Let x ∈ [a]. Then x R a.
Since a R b and R is transitive then x R b which means that x ∈ [b]. Thus,
[a] ⊆ [b]. Now interchange the letters a and b to show that [b] ⊆ [a]. Hence,
[a] = [b] which contradicts our assumption that [a] = [b]. This establishes
(ii). Thus, A/R is a partition of A.
The sets [a] deﬁned in the previous exercise are called the equivalence
classes of A given by the relation R. The element a in [a] is called a repre-
sentative of the equivalence class [a].
107.
18 EQUIVALENCE RELATIONS 107
Review Problems
Problem 18.1
Let X = {a, b, c}. Recall that P(X) is the power set of X. Deﬁne a binary
relation R on P(X) as follows:
A, B ∈ P(x), A R B ⇔ |A| = |B|.
a. Is {a, b}R{b, c}?
b. Is {a}R{a, b}?
c. Is {c}R{b}?
Problem 18.2
Let Σ = {a, b}. Then Σ4 is the set of all strings over Σ of length 4. Deﬁne a
relation R on Σ4 as follows:
s, t ∈ Σ4 , s R t ⇔ s has the same ﬁrst two characters as t.
a. Is abaa R abba?
b. Is aabb R bbaa?
c. Is aaaa R aaab?
Problem 18.3
Let A = {4, 5, 6} and B = {5, 6, 7} and deﬁne the binary relations R, S, and
T from A to B as follows:
(x, y) ∈ A × B, (x, y) ∈ R ⇔ x ≥ y.
(x, y) ∈ A × B, x S y ⇔ 2|(x − y).
T = {(4, 7), (6, 5), (6, 7)}.
a. Draw arrow diagrams for R, S, and T.
b. Indicate whether any of the relations S, R, or T are functions.
Problem 18.4
Let A = {3, 4, 5} and B = {4, 5, 6} and deﬁne the binary relation R as
follows:
(x, y) ∈ A × B, (x, y) ∈ R ⇔ x < y.
List the elements of the sets R and R−1 .
108.
108 RELATIONS AND FUNCTIONS
Problem 18.5
Let A = {2, 4} and B = {6, 8, 10} and deﬁne the binary relations R, S, and
T from A to B as follows:
(x, y) ∈ A × B, (x, y) ∈ R ⇔ x|y.
(x, y) ∈ A × B, x S y ⇔ y − 4 = x.
List the elements of A × B, R, S, R ∪ S, and R ∩ S.
Problem 18.6
Consider the binary relation on IR deﬁned as follows:
x, y ∈ R, x R y ⇔ x ≥ y.
Is R reﬂexive? symmetric? transitive?
Problem 18.7
Consider the binary relation on IR deﬁned as follows:
x, y ∈ R, x R y ⇔ xy ≥ 0.
Is R reﬂexive? symmetric? transitive?
Problem 18.8
Let Σ = {0, 1} and A = Σ∗ . Consider the binary relation on A deﬁned as
follows:
x, y ∈ A, x R y ⇔ |x| < |y|,
where |x| denotes the length of the string x. Is R reﬂexive? symmetric?
transitive?
Problem 18.9
Let A = ∅ and P(A) be the power set of A. Consider the binary relation on
P(A) deﬁned as follows:
X, Y ∈ P(A), X R Y ⇔ X ⊆ Y.
Is R reﬂexive? symmetric? transitive?
109.
18 EQUIVALENCE RELATIONS 109
Problem 18.10
Let E be the binary relation on Z deﬁned as follows:
a E b ⇔ m ≡ n (mod 2).
Show that E is an equivalence relation on Z and ﬁnd the diﬀerent equivalence
classes.
Problem 18.11
Let I be the binary relation on IR deﬁned as follows:
a I b ⇔ a − b ∈ Z.
Show that I is an equivalence relation on IR and ﬁnd the diﬀerent equivalence
classes.
Problem 18.12
Let A be the set all straight lines in the cartesian plane. Let || be the binary
relation on A deﬁned as follows:
l1 ||l2 ⇔ l1 is parallel to l2 .
Show that || is an equivalence relation on A and ﬁnd the diﬀerent equivalence
classes.
Problem 18.13
Let A = IN × IN. Deﬁne the binary relation R on A as follows:
(a, b) R (c, d) ⇔ a + d = b + c.
a. Show that R is reﬂexive.
b. Show that R is symmetric.
c. Show that R is transitive.
d. List ﬁve elements in [(1, 1)].
e. List ﬁve elements in [(3, 1)].
f. List ﬁve elements in [(1, 2)].
g. Describe the distinct equivalence classes of R.
Problem 18.14
Let R be a binary relation on a set A and suppose that R is symmetric and
transitive. Prove the following: If for every x ∈ A there is a y ∈ A such that
x R y then R is reﬂexive and hence an equivalence relation on A.
110.
110 RELATIONS AND FUNCTIONS
19 Partial Order Relations
A relation ≤ on a set A is called a partial order if ≤ is reﬂexive, antisym-
metric, and transitive. In this case we call A a poset.
Example 19.1
Show that the set Z of integers together with the relation of inequality ≤ is
a poset.
Solution.
≤ is reﬂexive: For all x ∈ Z we have x ≤ x since x = x.
≤ is antisymmetric: By the trichotomy law of real numbers, for a given pair
of numbers x and y only one of the following is true: x < y, x = y, or x > y.
So if x ≤ y and y ≤ x then we must have x = y.
≤ is transitive: By the transitivity property of < in IR if x < y and y < z
then x < z. Thus, if x ≤ y and y ≤ z then the deﬁnition of ≤ and the above
property imply that x ≤ z.
Example 19.2
Show that the relation a|b in IN∗ is a partial order relation.
Solution.
Reﬂexivity: Since a = 1 · a then a|a.
Antisymmetry: Suppose that a|b and b|a. Then there exist positive integers
k1 and k2 such that b = k1 a and a = k2 b. Hence, a = k1 k2 a which implies
that k1 k2 = 1. Since k1 , k2 ∈ IN∗ then we must have k1 = k2 = 1; that is,
a = b.
Transitivity: Suppose that a|b and b|c. Then there exist positive integers k1
and k2 such that b = k1 a and c = k2 b. Thus, c = k1 k2 a which means that
a|c.
Example 19.3
Let A be a collection of subsets. Let R be the relation deﬁned by
A R B ⇔ A ⊆ B.
Show that A is a poset.
111.
19 PARTIAL ORDER RELATIONS 111
Solution.
⊆ is reﬂexive: For any set X ∈ A, X ⊆ X.
⊆ is antisymmetric: By the deﬁnition of = if X ⊆ Y and Y ⊆ X then X = Y,
where X, Y ∈ A.
⊆ is transitive: We have seen in Chapter 3 that if X ⊆ Y and Y ⊆ Z then
X ⊆ Z.
To ﬁgure out which of two words comes ﬁrst in an English dictionary, one
compares their letters one by one from left to right. If all the letters have
been the same to a certain point and one word is runs out of letters, that word
comes ﬁrst in the dictionary. For example, play comes before playground. If
all the letters up to a certain point are the same and the next letters diﬀer,
then the word whose next letter is located earlier in the alphabet comes ﬁrst
in the dictionary. For example, playground comes before playmate. This type
of order relation is called lexicographic or dictionary order. A general de-
ﬁnition is the following:
Let Σ∗ be the set of words with letters from an ordered set Σ. Deﬁne the
relation ≤ on Σ∗ as follows: for all w, z ∈ Σ∗ , w ≤ z if and only if either
(a)z = wu for some u ∈ Σ∗ , or
(b)w = xu and z = xv where u, v ∈ Σ∗ such that the ﬁrst letter of u precedes
the ﬁrst letter of v in the ordering of Σ.
Then it can be shown that ≤ is a partial order relation on Σ∗ .
Example 19.4
Let Σ = {a, b} and suppose that Σ has the partial order relation R =
{(a, a), (a, b), (b, b)}. Let ≤ be the corresponding lexicographic order on Σ∗ .
Indicate which of the following statements are true.
a. aab ≤ aaba.
b. bbab ≤ bba.
c. ≤ aba.
d. aba ≤ abb.
e. bbab ≤ bbaa.
f. ababa ≤ ababaa.
g. bbaba ≤ bbabb.
112.
112 RELATIONS AND FUNCTIONS
Solution.
a. True since aaba = (aab)a.
b. False since bba ≤ bbab.
c. True since aba = aba.
d. True since aba = (ab)a, abb = (ab)b and a R b.
e. False since bbaa ≤ bbab.
f. True since ababaa = (ababa)a.
g. True since bbaba = (bbab)a, bbabb = (bbab)b and a R b.
Another simple pictorial representation of a partial order is the so called
Hasse diagram. The Hasse diagram of a partial order on the set A is a
drawing of the points of A and some of the arrows of the digraph of the order
relation. We assume that the directed edges of the digraph point upward.
There are rules to determine which arrows are drawn and which are omitted,
namely,
• omit all arrows that can be inferred from transitivity
• omit all loops
• draw arrows without ”heads”.
Example 19.5
Let A = {1, 2, 3, 9, 18} and the ”divides” relation on A. Draw the Hasse
diagram of this relation.
Solution.
The directed graph of the given relation is
The corresponding Hasse diagram is given by
113.
19 PARTIAL ORDER RELATIONS 113
Now, given the Hasse diagram of a partial order relation one can ﬁnd the
diagraph as follows:
• Reisert the direction markers on the arrows making all arrows point upward
• add loops at each vertex
• for each sequence of arrows from one point to a second point and from that
second point to a third point, add an arrow from the ﬁrst point to the third.
Example 19.6
Let A = {1, 2, 3, 4} be a poset. Find the directed graph corresponding to the
following Hasse diagram on A.
114.
114 RELATIONS AND FUNCTIONS
Solution.
Next, if A is a poset then we say that a and b are comparable if either
a ≤ b or b ≤ a. If every pair of elements of A are comparable then we call ≤
a totally ordered relation.
Example 19.7
Consider the ”divides” relation deﬁned on the set A = {5, 15, 30}. Prove that
this relation is a total order on A.
Solution.
The facts that ”divides” relation is a partial order relation is easy to verify.
Since 5|15, 5|30, and 15|30 then any pair of elements in A are comparable.
Thus, the ”divides” relation is a total order relation on A.
Example 19.8
Show that the ”divides” relation on IN∗ is not a total partial order.
Solution.
A counterexample of two noncomparable numbers are 2 and 3. Since 2 does
not divide 3 and 3 does not divide 2.
115.
19 PARTIAL ORDER RELATIONS 115
Review Problems
Problem 19.1
Let Σ = {a, b} and let Σ∗ be the set of all strings over Σ. Deﬁne the relation
R on Σ∗ as follows: for all s, t ∈ Σ∗ ,
s R t ⇔ l(s) ≤ l(t),
where l(x) denotes the length of the word x. Is R antisymmetric? Prove or
give a counterexample.
Problem 19.2
Deﬁne a relation R on Z as follows: for all m, n ∈ Z
m R n ⇔ m + n is even.
Is R a partial order? Prove or give a counterexample.
Problem 19.3
Deﬁne a relation R on IR as follows: for all m, n ∈ IR
m R n ⇔ m2 ≤ n2 .
Is R a partial order? Prove or give a counterexample.
Problem 19.4
Let S = {0, 1} and consider the partial order relation R deﬁned on S × S as
follows: for all ordered pairs (a, b) and (c, d) in S × S
(a, b) R (c, d) ⇔ a ≤ c and b ≤ d.
Draw the Hasse diagram for R.
Problem 19.5
Consider the ”divides” relation deﬁned on the set A = {1, 2, 22 , · · · , 2n },
where n is a nonnegative integer.
a. Prove that this relation is a total order on A.
b. Draw the Hasse diagram for this relation when n = 3.
116.
116 RELATIONS AND FUNCTIONS
20 Functions: Deﬁnitions and Examples
A function is a special case of a relation. A function f from a set A to a
set B is a relation from A to B such that for every x ∈ A there is a unique
y ∈ B such that (x, y) ∈ f. For (x, y) ∈ f we use the notation y = f (x). We
call y the image of x under f. The set A is called the domain of f whereas
B is called the codomain. The collection of all images of f is called the
range of f.
Example 20.1
Show that the relation f = {(1, a), (2, b), (3, a)} deﬁnes a function from A =
{1, 2, 3} to B = {a, b, c}. Find its range.
Solution.
Since every element of A has a unique image then f is a function. Its range
consists of the elements a and b.
Example 20.2
Show that the relation f = {(1, a), (2, b), (3, c), (1, b)} does not deﬁne a func-
tion from A = {1, 2, 3} to B = {a, b, c}.
Solution.
Indeed, since 1 has two images in B then f is not a function.
Example 20.3
A sequence of elements of a set A is a function from IN∗ to A. We write
(an ) and we call an the nth term of the sequence.
a. Deﬁne the sequence an = n, n ≥ 1. Compute n ak .k=1
b. Deﬁne the sequence an = n2 . Compute the sum n ak . k=1
Solution.
a. Let Sn = n ak . Then write Sn in two diﬀerent ways, namely, Sn =
k=1
1 + 2 + · · · + n and Sn = n + (n − 1) + · · · + 1. Adding, we obtain 2Sn =
(n + 1) + (n + 1) + · · · + (n + 1) = n(n + 1). Thus, Sn = n(n+1) .
2
b. First note that (n + 1)3 − n3 = 3n2 + 3n + 1. From this we obtain the
following chain of equalities:
23 − 13 = 3(1)2 + 3(1) + 1
33 − 23 = 3(2)2 + 3(2) + 1
.
.
.
(n + 1)3 − n3 = 3n2 + 3n + 1
117.
20 FUNCTIONS: DEFINITIONS AND EXAMPLES 117
Adding these equalities we ﬁnd
n n
2
3 k +3 k + n = (n + 1)3 − 1.
k=1 k=1
Using a. we ﬁnd
n
3n(n + 1)
3 k2 + + n = n3 + 3n2 + 3n.
k=1
2
A simple arithmetic shows that
n
n(n + 1)(2n + 1)
k2 = .
k=1
6
Example 20.4
Let A = {a, b, c}. Deﬁne the function f : P(A) → IN by f (X) = |X|. Find
the range of f.
Solution.
By applying f to each member of P(A) we ﬁnd Range(f ) = {0, 1, 2, 3}.
Example 20.5
Consider the alphabet Σ = {a, b} and the function f : Σ∗ → Z deﬁned as
follows: for any string s ∈ Σ∗
f (s) = the number of a s in s.
Find f ( ), f (ababb), and f (bbbaa).
Solution.
f ( ) = 0, f (ababb) = 2, and f (bbbaa) = 2.
Example 20.6 (Equality of Functions)
Two functions f and g deﬁned on the same domain D are said to be equal if
and only if f (x) = g(x) for all x ∈√ Show that the functions f, g : IR → IR
D.
deﬁned by f (x) = |x| and g(x) = x2 are equal.
Solution. √
A simple argument by the method of proof by cases shows that x2 = |x|.
118.
118 RELATIONS AND FUNCTIONS
Example 20.7 (Hamming distance function)
Let Σ = {0, 1} and Σn be the set of all strings of 0’s and 1’s of length n.
Deﬁne the function H : Σn × Σn → IN as follows: for any (s, t) ∈ Σn × Σn
H(s, t) = number of positions in which s and t have dif f erent values.
For the case n = 5, ﬁnd H(00101, 01110) and H(10001, 01111).
Solution.
H(00101, 01110) = 3 and H(10001, 01111) = 4.
Example 20.8 (Boolean functions)
An n-place Boolean function f is a function from the Cartesian product
{0, 1}n to {0, 1}. Consider the 3-place Boolean function f : {0, 1}3 → {0, 1}
deﬁned by
f (x1 , x2 , x3 ) = (x1 + x2 + x3 ) mod 2.
Describe f using an input/output table.
Solution.
x1 x2 x3 f (x1 , x2 , x3 )
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 1
0 0 1 1
0 0 0 0
Example 20.9 (Encoding and Decoding functions)
Let Σ = {0, 1} and Σ∗ be the set of all strings of 0’s and 1’s. Let L be the
set of all strings over Σ that consist of consecutive triples of identical bits.
Thus, 111000 ∈ L. A message consisting of 0’s and 1’s is encoded by writing
each bit in it three times. The encoded message is decoded by replacing each
section of three identical bits by the one bit to which all three are equal.
We deﬁne the encoding function E : Σ∗ → L by
E(s) = the string obtained f rom s by replacing each bit of s
by the same bit written three times
119.
20 FUNCTIONS: DEFINITIONS AND EXAMPLES 119
and we deﬁne the decoding function D : L → Σ∗ by
D(s) = the string obtained f rom s by replacing consecutive triple of bits
of s by a single copy of that bit.
Find E(0110) and D(111111000111).
Solution.
We have E(0110) = 000111111000 and D(111111000111) = 1101.
Now, let A and B be subsets of IR. A function f : A → B is called a
real-valued function of a real variable. In this case, each ordered pair
(x, f (x)) can be represented by a point in the Cartesian plane. The collection
of all such points is called the graph of f.
Example 20.10
Consider the power function fa (x) = xa , where a, x ∈ IR+ ∪ {0}. Graph on
the same Cartesian plane the functions f0 (x), f1 (x), f 1 (x), and f2 (x).
2
Solution.
Example 20.11
Graph the functions f (x) = x and g(x) = x on the closed interval [−4, 4].
120.
120 RELATIONS AND FUNCTIONS
Solution.
Example 20.12 √
Graph the function f : IN → IR deﬁned by f (n) = n.
Solution.
121.
20 FUNCTIONS: DEFINITIONS AND EXAMPLES 121
Example 20.13
Let Df be the domain of a function f and S ⊆ Df . We say that f is increas-
ing on S if and only if, for all x1 , x2 ∈ S, if x1 < x2 then f (x1 ) < f (x2 ).
Show that the function f : IR → IR deﬁned by f (x) = 2x − 3 is increasing
on IR.
Solution.
Indeed, for any real numbers x1 and x2 such that x1 < x2 , we have 2x1 − 3 <
2x2 − 3. That is, f (x1 ) < f (x2 ) so that f is increasing.
Example 20.14
Let Df be the domain of a function f and S ⊆ Df . We say that f is de-
creasing on S if and only if, for all x1 , x2 ∈ S, if x1 < x2 then f (x1 ) > f (x2 ).
Show that the function f : IR → IR deﬁned by f (x) = x+2 is decreasing on
x+1
(−∞, −1) and (−1, ∞).
Solution.
Indeed, for any real numbers x1 , x2 ∈ (−∞, −1) or x1 , x2 ∈ (−1, ∞) such that
x1 < x2 , we have (x1 + 1)(x2 + 1) > 0. This implies, that f (x1 ) − f (x2 ) =
x2 −x1
(x1 +1)(x2 +1)
> 0. Thus, f is decreasing on the given intervals.
122.
122 RELATIONS AND FUNCTIONS
Review Problems
Problem 20.1
2x3 +2x2
Let f, g : IR → IR be the functions f (x) = 2x and g(x) = x2 +1
. Show that
f = g.
Problem 20.2
Let H, K : IR → IR be the functions H(x) = x + 1 and K(x) = x . Does
H = K? Explain.
Problem 20.3
Find functions deﬁned on the set of nonnegative integers that deﬁne the
sequences whose ﬁrst six terms are given below.
a. 1, − 1 , 1 , − 7 , 1 , − 11 .
3 5
1
9
1
b. 0, −2, 4, −6, 8, −10.
Problem 20.4
Let A = {1, 2, 3, 4, 5} and let F : P(A) → Z be deﬁned as follows:
0 if X has an even number of elements
F (X) =
1 if X has an odd number of elements
Find the following
a. F ({1, 3, 4})
b. F (∅).
c. F ({2, 3}).
d. F ({2, 3, 4, 5}).
Problem 20.5
Let Σ = {a, b} and Σ∗ be the set of all strings over Σ.
a. Deﬁne f : Σ∗ → Z as follows:
the number of b s to the lef t of the lef t − most a in s
f (s) =
0 if s contains no a s
Find f (aba), f (bbab), and f (b). What is the range of f ?
b. Deﬁne g : Σ∗ → Σ∗ as follows:
g(s) = the string obtained by writing the characters of s in reverse order.
Find g(aba), g(bbab), and g(b). What is the range of g?
123.
20 FUNCTIONS: DEFINITIONS AND EXAMPLES 123
Problem 20.6
Let E and D be the encoding and decoding functions.
a. Find E(0110) and D(111111000111).
b. Find E(1010) and D(000000111111).
Problem 20.7
Let H denote the Hamming distance function on Σ5 .
a. Find H(10101, 00011).
b. Find H(00110, 10111).
Problem 20.8
Consider the three-place Boolean function f : {0, 1}3 → {0, 1} deﬁned as
follows:
f (x1 , x2 , x3 ) = (3x1 + x2 + 2x3 ) mod 2
a. Find f (1, 1, 1) and f (0, 1, 1).
b. Describe f using an input/output table.
Problem 20.9
Draw the graphs of the power functions f 1 (x) and f 1 (x) on the same set of
3 4
1 1
axes. When, 0 < x < 1, which is greater: x 3 or x 4 ? When x > 1, which is
1 1
greater s 3 or x 4 ?
Problem 20.10
Graph the function f (x) = x − x on the interval (−∞, ∞).
Problem 20.11
Graph the function f (x) = x − x on the inerval (−∞, ∞).
Problem 20.12
n
Graph the function h : IN → IR deﬁned by h(n) = 2
.
Problem 20.13
x−1
Let k : IR → IR be the function deﬁned by the formula k(x) = x
for all
nonzero real numbers x.
a. Show that k is increasing on (0, ∞).
b. Is k increasing or decreasing on (−∞, 0)? Prove your answer.
124.
124 RELATIONS AND FUNCTIONS
21 Bijective and Inverse Functions
Let f : A → B be a function. We say that f is injective or one-to-one if
and only if for all x, y ∈ A, if f (x) = f (y) then x = y. Using the concept of
contrapositive, a function f is injective if and only if for all x, y ∈ A, if x = y
then f (x) = f (y). Taking the negation of this last conditional implication
we see that f is not injective if and only if there exist two distinct elements
a and b of A such that f (a) = f (b).
Example 21.1
a. Show that the identity function IA on a set A is injective.
b. Show that the fucntion f : Z → Z deﬁned by f (n) = n2 is not injective.
Solution.
a. Let x, y ∈ A. If IA (x) = IA (y) then x = y by the deﬁnition of IA . This
shows that IA is injective.
b. Since 12 = (−1)2 and 1 = −1, f is not injective.
Example 21.2 (Hash Functions)
Let m > 1 be a positive integer . Show that the function h : Z → Z deﬁned
by h(n) = n mod m is not injective.
Solution.
Indeed, since m > 1 then 2m + 1 = m + 1 and h(m + 1) = h(2m + 1) = 1.
So h is not injective.
Example 21.3
Show that if f : IR → IR is increasing then f is one-to-one.
Solution.
Suppose that x1 = x2 . Then without lost of generality we can assume that
x1 < x2 . Since f is increasing then f (x1 ) < f (x2 ). That is, f (x1 ) = f (x2 ).
Hence, f is one-to-one.
Example 21.4
Show that the composition of two injective functions is also injective.
Solution.
Let f : A → B and g : B → C be two injective functions. We will show that
g◦f : A → C is also injective. Indeed, suppose that (g◦f )(x1 ) = (g◦f )(x2 ) for
125.
21 BIJECTIVE AND INVERSE FUNCTIONS 125
x1 , x2 ∈ A. Then g(f (x1 )) = g(f (x2 )). Since g is injective then f (x1 ) = f (x2 ).
Now since f is injective then x1 = x2 . This completes the proof that g ◦ f is
injective.
Now, for any function f : A → B we have Range(f ) ⊆ B. If equality holds
then we say that f is surjective or onto. It follows from this deﬁnition that
a function f is surjective if and only if for each y ∈ B there is an x ∈ A such
that f (x) = y. By taking the negation of this we see that f is not onto if
there is a y ∈ B such that f (x) = y for all x ∈ A.
Example 21.5
a. Show that the function f : IR → IR deﬁned by f (x) = 3x − 5 is surjective.
b. Show that the function f : Z → Z deﬁned by f (n) = 3n − 5 is not
surjective.
Solution.
a. Let y ∈ IR. Is there an x ∈ IR such that f (x) = y? That is, 3x − 5 = y.
But solving for x we ﬁnd x = y+5 ∈ IR and f (x) = y. Thus, f is onto.
3
b. Take m = 3. If f is onto then there should be an n ∈ Z such that f (n) = 3.
That is, 3n − 5 = 3. Solving for n we ﬁnd n = 8 which is not an integer.
3
Hence, f is not onto.
Example 21.6 (Projection Functions)
Let A and B be two nonempty sets. The functions prA : A × B → A deﬁned
by prA (a, b) = a and prB : A × B → B deﬁned by prB (a, b) = b are called
projection functions. Show that prA and prB are surjective functions.
Solution.
We prove that prA is surjective. Indeed, let a ∈ A. Since B is not empty
then there is a b ∈ B. But then (a, b) ∈ A × B and prA (a, b) = a. Hence, prA
is surjective. The proof that prB is surjective is similar.
Example 21.7
Show that the composition of two surjective functions is also surjective.
Solution.
Let f : A → B and g : B → C, where Range(f ) ⊆ C, be two surjective
functions. We will show that g ◦ f : A → D is also surjective. Indeed, let
z ∈ D. Since g is surjective then there is a y ∈ B such that g(y) = z. Since f
126.
126 RELATIONS AND FUNCTIONS
is surjective then there is an x ∈ A such that f (x) = y. Thus, g(f (x)) = z.
This shows that g ◦ f is surjective.
Now, we say that a function f is bijective or one-to-one correspondence
if and only if f is both injective and surjective. A bijective function on a set
A is called a permutation.
Example 21.8
a. Show that the function f : IR → IR deﬁned by f (x) = 3x − 5 is a bijective
function.
b. Show that the function f : IR → IR deﬁned by f (x) = x2 is not bijective.
Solution.
a. First we show that f is injective. Indeed, suppose that f (x1 ) = f (x2 ).
Then 3x1 − 5 = 3x2 − 5 and this implies that x1 = x2 . Hence, f is injective.
f is surjective by Exercise 309 (a).
b. f is not injective since f (−1) = f (1) but −1 = 1. Hence, f is not bijective.
Example 21.9
Show that the composition of two bijective functions is also bijective.
Solution.
This follows from Example 21.4 and Example 21.7
Theorem 21.1
Let f : X → Y be a bijective function. Then there is a function f −1 : Y → X
with the following properties:
a. f −1 (y) = x if and only if f (x) = y.
b. f −1 ◦ f = IX and f ◦ f −1 = IY where IX denotes the identity function on
X.
c. f −1 is bijective.
Proof.
For each y ∈ Y there is a unique x ∈ X such that f (x) = y since f is
bijective. Thus, we can deﬁne a function f −1 : Y → X by f −1 (y) = x where
f (x) = y.
a. Follows from the deﬁnition of f −1 .
127.
21 BIJECTIVE AND INVERSE FUNCTIONS 127
b. Indeed, let x ∈ X such that f (x) = y. Then f −1 (y) = x and (f −1 ◦f )(x) =
f −1 (f (x)) = f −1 (y) = x = IX (x). Since x was arbitrary then f −1 ◦ f = IX .
The proof that f ◦ f −1 = IY is similar.
c. We show ﬁrst that f −1 is injective. Indeed, suppose f −1 (y1 ) = f −1 (y2 ).
Then f (f −1 (y1 )) = f (f −1 (y2 )); that is, (f ◦ f −1 )(y1 ) = (f ◦ f −1 )(y2 ). By b.
we have IY (y1 ) = IY (y2 ). From the deﬁnition of IY we obtain y1 = y2 . Hence,
f −1 is injective. We next show that f −1 is surjective. Indeed, let y ∈ Y .
Since f is onto there is a unique x ∈ X such that f (x) = y. By the deﬁntion
of f −1 , f −1 (y) = x. Thus, for every element y ∈ Y there is an element x ∈ X
such that f −1 (y) = x. This says that f −1 is surjective and completes a proof
of the theorem
Example 21.10
Show that f : IR → IR deﬁned by f (x) = 3x − 5 is bijective and ﬁnd a
formula for its inverse function.
Solution.
We have already proved that f is bijective. We will just ﬁnd the formula
for its inverse function f −1 . Indeed, if y ∈ Y we want to ﬁnd x ∈ X such
that f −1 (y) = x, or equivalently, f (x) = y. This implies that 3x − 5 = y and
solving for x we ﬁnd x = y+5 . Thus, f −1 (y) = y+5
3 3
128.
128 RELATIONS AND FUNCTIONS
Review Problems
Problem 21.1
a. Deﬁne g : Z → Z by g(n) = 3n − 2.
(i) Is g one-to-one? Prove or give a counterexample.
(ii) Is g onto? Prove or give a counterexample.
b. Deﬁne G : IR → IR by G(x) = 3x − 2. Is G onto? Prove or give a
counterexample.
Problem 21.2
x+1
Determine whether the function f : IR → IR given by f (x) = x
is one-to-
one or not.
Problem 21.3
x
Determine whether the function f : IR → IR given by f (x) = x2 +1
is one-to-
one or not.
Problem 21.4
Let f : IR → Z be the ﬂoor function f (x) = x .
a. Is f one-to-one? Prove or give a counterexample.
b. Is f onto? Prove or give a counterexample.
Problem 21.5
Let Σ = {0, 1} and let l : Σ∗ → IN denote the length function.
a. Is l one-to-one? Prove or give a counterexample.
b. Is l onto? Prove or give a counterexample.
Problem 21.6
If : IR → IR and g : IR → IR are one-to-one functions, is f +g also one-to-one?
Justify your answer.
Problem 21.7
Deﬁne F : P{a, b, c} → IN to be the number of elements of a subset of
P{a, b, c}.
a. Is F one-to-one? Prove or give a counterexample.
b. Is F onto? Prove or give a counterexample.
Problem 21.8
If : IR → IR and g : IR → IR are onto functions, is f + g also onto? Justify
your answer.
129.
21 BIJECTIVE AND INVERSE FUNCTIONS 129
Problem 21.9
Let Σ = {a, b} and let l : Σ∗ → IN be the length function. Let f : IN →
{0, 1, 2} be the hash function f (n) = n mod 3. Find (f ◦l)(abaa), (f ◦l)(baaab),
and (f ◦ l)(aaa).
Problem 21.10
y−2
Show that the function F −1 : IR → IR given by F −1 (y) = 3
is the inverse
of the function F (x) = 3x + 2.
Problem 21.11
If f : X → Y and g : Y → Z are functions and g ◦ f : X → Z is one-to-one,
must both f and g be one-to-one? Prove or give a counterexample.
Problem 21.12
If f : X → Y and g : Y → Z are functions and g ◦ f : X → Z is onto, must
both f and g be onto? Prove or give a counterexample.
Problem 21.13
If f : X → Y and g : Y → Z are functions and g ◦ f : X → Z is one-to-one,
must f be one-to-one? Prove or give a counterexample.
Problem 21.14
If f : X → Y and g : Y → Z are functions and g ◦ f : X → Z is onto, must
g be onto? Prove or give a counterexample.
Problem 21.15
Let f : W → X, g : X → Y and h : Y → Z be functions. Must h ◦ (g ◦ f ) =
(h ◦ g) ◦ f ? Prove or give a counterexample.
Problem 21.16
Let f : X → Y and g : Y → Z be two bijective functions. Show that (g ◦f )−1
exists and (g ◦ f )−1 = f −1 ◦ g −1 .
130.
130 RELATIONS AND FUNCTIONS
22 Recursion
A recurrence relation for a sequence a0 , a1 , · · · is a relation that deﬁnes
an in terms of a0 , a1 , · · · , an−1 . The formula relating an to earlier values in
the sequence is called the generating rule. The assignment of a value to
one of the a’s is called an initial condition.
Example 22.1
The Fibonacci sequence
1, 1, 2, 3, 5, · · ·
is a sequence in which every number after the ﬁrst two is the sum of the
preceding two numbers. Find the generating rule and the initial conditions.
Solution.
The initial conditions are a0 = a1 = 1 and the generating rule is an =
an−1 + an−2 , n ≥ 2.
Example 22.2
Let n ≥ 0 and ﬁnd the number sn of words from the alphabet Σ = {0, 1} of
length n not containing the pattern 11 as a subword.
Solution.
Clearly, s0 = 1(empty word) and s1 = 2. We will ﬁnd a recurrence relation
for sn , n ≥ 2. Any word of length n with letters from Σ begins with either
0 or 1. If the word begins with 0, then the remaining n − 1 letters can be
any sequence of 0’s or 1’s except that 11 cannot happen. If the word begins
with 1 then the next letter must be 0 since 11 can not happen; the remaining
n − 2 letters can be any sequence of 0’s and 1’s with the exception that 11
is not allowed. Thus the above two categories form a partition of the set of
all words of length n with letters from Σ and that do not contain 11. This
implies the recurrence relation
sn = sn−1 + sn−2 , n ≥ 2
A solution to a recurrence relation is an explicit formula for an in terms of
n.
The most basic method for ﬁnding the solution of a sequence deﬁned recur-
sively is by using iteration. The iteration method consists of starting with
the initial values of the sequence and then calculate successive terms of the
131.
22 RECURSION 131
sequence until a pattern is observed. At that point one guesses an explicit
formula for the sequence and then uses mathematical induction to prove its
validity.
Example 22.3
Find a solution for the recurrence relation
a0 =1
an = an−1 + 2, n ≥ 1
Solution.
Listing the ﬁrst ﬁve terms of the sequence one ﬁnds
a0 = 1
a1 = 1+2
a2 = 1+4
a3 = 1+6
a4 = 1+8
Hence, a guess is an = 2n + 1, n ≥ 0. It remains to show that this formula is
valid by using mathematical induction.
Basis of induction: For n = 0, a0 = 1 = 2(0) + 1.
Induction hypothesis: Suppose that an = 2n + 1.
Induction step: We must show that an+1 = 2(n + 1) + 1. By the deﬁnition of
an+1 we have an+1 = an + 2 = 2n + 1 + 2 = 2(n + 1) + 1.
Example 22.4
Consider the arithmetic sequence
an = an−1 + d, n ≥ 1
where a0 is the initial value. Find an explicit formula for an .
Solution. Listing the ﬁrst four terms of the sequence after a0 we ﬁnd
a1 = a0 + d
a2 = a0 + 2d
a3 = a0 + 3d
a4 = a0 + 4d
132.
132 RELATIONS AND FUNCTIONS
Hence, a guess is an = a0 + nd. Next, we prove the validity of this formula
by induction.
Basis of induction: For n = 0, a0 = a0 + (0)d.
Induction hypothesis: Suppose that an = a0 + nd.
Induction step: We must show that an+1 = a0 + (n + 1)d. By the deﬁnition
of an+1 we have an+1 = an + d = a0 + nd + d = a0 + (n + 1)d.
Example 22.5
Consider the geometric sequence
an = ran−1 , n ≥ 1
where a0 is the initial value. Find an explicit formula for an .
Solution.
Listing the ﬁrst four terms of the sequence after a0 we ﬁnd
a1 = ra0
a2 = r 2 a0
a3 = r 3 a0
a4 = r 4 a0
Hence, a guess is an = rn a0 . Next, we prove the validity of this formula by
induction.
Basis of induction: For n = 0, a0 = r0 a0 .
Induction hypothesis: Suppose that an = rn a0 .
Induction step: We must show that an+1 = rn+1 a0 . By the deﬁnition of an+1
we have an+1 = ran = r(rn a0 ) = rn+1 a0 .
Example 22.6
Find a solution to the recurrence relation
a0 =0
an = an−1 + (n − 1), n ≥ 1
Solution.
Writing the ﬁrst ﬁve terms of the sequence we ﬁnd
a0 = 0
a1 = 0
a2 = 0+1
a3 = 0+1+2
a4 = 0+1+2+3
133.
22 RECURSION 133
A guessing formula is that
n(n − 1)
an = 0 + 1 + 2 + · · · + (n − 1) = .
2
We next show that the formula is valid by using induction on n ≥ 0.
Basis of induction: a0 = 0 = 0(0−1) .
2
Induction hypothesis: Suppose that an = n(n−1) .
2
Induction step: We must show that an+1 = n(n+1) . Indeed,
2
an+1 = an + n
n(n−1)
= 2
+n
n(n+1)
= 2
Example 22.7
Consider the recurrence relation
a0 =1
an = 2an−1 + n, n ≥ 1
Is it true that an = 2n + n is a solution to the given recurrence relation?
Solution.
If so then we must be able to prove its validity by mathematical induction.
Basis of induction: a0 = 20 + 1.
Induction hypothesis: Suppose that an = 2n + n.
Induction step: We must show that an+1 = 2n+1 +(n+1). If this is so then we
will have 2n+1 + (n + 1) = 2an + n = 2n+1 + 2n + n + 1. But this would imply
that n = 0 which contradicts the fact that n is any nonnegative integer.
Example 22.8
Deﬁne a sequence, a1 , a2 , · · · , recursively as follows:
a1 =1
an = 2 · a n , n ≥ 2
2
a. Use iteration to guess an explicit formula for this sequence.
b. Use induction to prove the validity of the formula found in a.
134.
134 RELATIONS AND FUNCTIONS
Solution.
Computing the ﬁrst few terms of the sequence we ﬁnd
a1 = 1
a2 = 2
a3 = 2
a4 = 4
a5 = 4
a6 = 4
a7 = 4
a8 = · · · = a15 = 8
Hence, for 2i ≤ n < 2i+1 , an = 2i . Moreover, i ≤ log2 n < i + 1 so that
i = log2 n and a formula for an is
log2 n
an = 2 , n ≥ 1.
b. We prove the above formula by mathematical induction.
Basis of induction: For n = 1, a1 = 1 = 2 log2 1 .
Induction hypothesis: Suppose that an = 2 log2 n .
Induction step: We must show that an+1 = 2 log2 (n+1) . Indeed, for n odd
(i.e. n + 1 even) we have
an+1 = 2 · a n+1
2
= 2 · a n+1
2
n+1
= 2 · 2 log2 2
= 2 log2 (n+1)−1 +1
= 2 log2 (n+1) −1+1
= 2 log2 (n+1)
A similar argument holds when n is even.
When iteration does not apply, other methods are available for ﬁnding ex-
plicit formulas for special classes of recursively deﬁned sequences. The method
explained below works for sequences of the form
an = Aan−1 + Ban−2 (22.1)
135.
22 RECURSION 135
where n is greater than or equal to some ﬁxed nonnegative integer k and A
and B are real numbers with B = 0. Such an equation is called a second-
order linear homogeneous recurrence relation with constant coeﬃ-
cients.
Example 22.9
Does the Fibonacci sequence satisfy a second-order linear homogeneous re-
lation with constant coeﬃcients?
Solution.
Recall that the Fibonacci sequence is deﬁned recursively by an = an−1 + an−2
for n ≥ 2 and a0 = a1 = 1. Thus, an satisﬁes a second-order linear homoge-
neous relation with A = B = 1.
The following theorem gives a technique for ﬁnding solutions to (22.1).
Theorem 22.1
Equation (22.1) is satisﬁed by the sequence 1, t, t2 , · · · , tn , · · · where t = 0 if
and only if t is a solution to the characteristic equation
t2 − At − B = 0 (22.2)
Proof.
(=⇒): Suppose that t is a nonzero real number such that the sequence
1, t, t2 , · · · satisﬁes (22.1). We will show that t satisﬁes the equation t2 −
At − B = 0. Indeed, for n ≥ k we have
tn = Atn−1 + Btn−2 .
Since t = 0 we can divide through by tn−2 and obtain t2 − At − B = 0.
(⇐=) : Suppose that t is a nonzero real number such that t2 − At − B = 0.
Multiply both sides of this equation by tn−2 to obtain
tn = Atn−1 + Btn−2 .
This says that the sequence 1, t, t2 , · · · satisﬁes (22.1)
Example 22.10
Consider the recurrence relation
an = an−1 + 2an−2 , n ≥ 2.
Find two sequences that satisfy the given generating rule and have the form
1, t, t2 , · · · .
136.
136 RELATIONS AND FUNCTIONS
Solution.
According to the previous theorem t must satisfy the characteristic equation
t2 − t − 2 = 0.
Solving for t we ﬁnd t = 2 or t = −1. So the two solutions to the given
recurrence sequence are 1, 2, 22 , · · · , 2n , · · · and 1, −1, · · · , (−1)n , · · ·
Are there other solutions then the ones provided by Theorem 34.1? The
answer is yes according to the following theorem.
Theorem 22.2
If sn and tn are solutions to (22.1) then for any real numbers C and D the
sequence
an = Csn + Dtn , n ≥ 0
is also a solution.
Proof.
Since sn and tn are solutions to (22.1), for n ≥ 2 we have
sn = Asn−1 + Bsn−2
tn = Atn−1 + Btn−2
Therefore,
Aan−1 + Ban−2 = A(Csn−1 + Dtn−1 ) + B(Csn−2 + Dtn−2 )
= C(Asn−1 + Bsn−2 ) + D(Atn−1 + Btn−2 )
= Csn + Dtn = an
so that an satisﬁes (22.1)
Example 22.11
Find a solution to the recurrence relation
a0 = 1, a1 = 8
an = an−1 + 2an−2 , n ≥ 2.
Solution.
By the previous theorem and Example 22.10, an = C2n + D(−1)n , n ≥ 2 is
a solution to the recurrence relation
an = an−1 + 2an−2 .
137.
22 RECURSION 137
If an satisﬁes the system then we must have
a0 = C20 + D(−1)0
a1 = C21 + D(−1)1
This yields the system
C +D = 1
2C − D = 8
Solving this system to ﬁnd C = 3 and D = −2. Hence, an = 3 · 2n − 2(−1)n .
Example 22.12
Find an explicit formula for the Fibonacci sequence
a0 = a1 = 1
an = an−1 + an−2
Solution.
The roots of the characteristic equation
t2 − t − 1 = 0
√ √
1− 5 1+ 5
are t = 2
and t = 2
. Thus,
√ √
1+ 5 n 1− 5 n
an = C( ) + D( )
2 2
is a solution to
an = an−1 + an−2 .
Using the values of a0 and a1 we obtain the system
√
C +D √ = 1
C( 1+2 5 ) 1− 5
+ D( 2 ) = 1.
Solving this system to obtain
√ √
1+ 5 1− 5
C= √ and D = − √ .
2 5 2 5
Hence, √ √
1 1 + 5 n+1 1 1 − 5 n+1
an = √ ( ) −√ ( )
5 2 5 2
Next, we discuss the case when the characteristic equation has a single root.
138.
138 RELATIONS AND FUNCTIONS
Theorem 22.3
Let A and B be real numbers and suppose that the characteristic equation
t2 − At − B = 0
has a single root r. Then the sequences {1, r, r2 , · · · } and {0, r, 2r2 , 3r3 , · · · , nrn , · · · }
both satisfy the recurrence relation
an = Aan−1 + Ban−2 .
Proof.
Since r is a root to the characteristic equation, the sequence {1, r, r2 , · · · } is
a solution to the recurrence relation
an = Aan−1 + Ban−2 .
Now, since r is the only solution to the characteristic equation we have
(t − r)2 = t2 − At − B.
This implies that A = 2r and B = −r2 . Let sn = nrn , n ≥ 0. Then
Asn−1 + Bsn−2 = A(n − 1)rn−1 + B(n − 2)rn−2
= 2r(n − 1)rn−1 − r2 (n − 2)rn−2
= 2(n − 1)rn − (n − 2)rn
= nrn = sn
So sn is a solution to an = Aan−1 + Ban−2 .
Example 22.13
Find an explicit formula for
a0 = 1, a1 = 3
an = 4an−1 − 4an−2 , n ≥ 2
Solution.
Solving the characteristic equation
t2 − 4t + 4 = 0
we ﬁnd the single root r = 2. Thus,
an = C2n + Dn2n
139.
22 RECURSION 139
is a solution to the equation an = 4an−1 − 4an−2 . Since a0 = 1 and a1 = 3
then we obtain the following system of equations:
C = 1
2C + 2D = 3
Solving this system to obtain C = 1 and D = 1 . Hence, an = 2n + n 2n .
2 2
Example 22.14
Let A1 , A2 , · · · , An be subsets of a set S.
a. Give a recursion deﬁnition for ∪n Ai .i=1
b. Give a recursion deﬁnition for ∩n Ai .i=1
Solution.
a. ∪1 Ai = A1 and ∪n Ai = (∪n−1 Ai ) ∪ An , n ≥ 2.
i=1 i=1 i=1
n−1
b. ∩1 Ai = A1 and ∩n Ai = (∩i=1 Ai ) ∩ An , n ≥ 2.
i=1 i=1
Example 22.15
Use mathematical induction to prove the following generalized De Morgan’s
law.
(∪n Ai )c = ∩n Ac
i=1 i=1 i
Solution.
Basis of induction: (∪1 Ai )c = Ac = ∩1 Ac .
i=1 i i=1 i
Induction hypothesis: Suppose that (∪n Ai )c = ∩n Ac .
i=1 i=1 i
Induction step: We must show that (∪n+1 Ai )c = ∩n+1 Ac . Indeed,
i=1 i=1 i
(∪n+1 Ai )c = ((∪n Ai ) ∪ An+1 )c
i=1 i=1
= (∪n Ai )c ) ∩ Ac
i=1 n+1
= (∩n Ac ) ∩ Ac
i=1 i n+1
= ∩n+1 Ac
i=1 i
Example 22.16
Let a1 , a2 , · · · , an be numbers.
a. Give a recursion deﬁnition for n ai .
i=1
b. Give a recursion deﬁnition for Πn ai .
i=1
Solution.
a. 1 ai = a1 and n ai = ( i=1 ai ) + an , n ≥ 2.
i=1 i=1
n−1
n−1
b. Π1 ai = a1 and Πn ai = (Πi=1 ai ) · an , n ≥ 2.
i=1 i=1
140.
140 RELATIONS AND FUNCTIONS
Example 22.17
A function is said to be deﬁned recursively or to be a recursive function
if its rule of deﬁnition refers to itself. Deﬁne the factorial function recursively.
Solution.
f (0) =1
f (n) = nf (n − 1), n ≥ 1.
Example 22.18
Let G : IN → Z be the relation given by
1, if n = 1
n
G(n) = 1 + G( 2 ), if n is even
G(3n − 1), if n > 1 is odd
Show that G is not a function.
Solution.
Assume that G is a function so that G(5) exists. Listing the ﬁrst ﬁve values
of G we ﬁnd
G(1) = 1
G(2) = 2
G(3) = G(8) = 1 + G(4) = 2 + G(2) = 4
G(4) = 1 + G(2) = 3
G(5) = G(14) = 1 + G(7)
= 1 + G(20)
= 2 + G(10)
= 3 + G(5)
But the last equality implies that 0 = 3 which is impossible. Hence, G does
not deﬁne a function.
141.
22 RECURSION 141
Review Problems
Problem 22.1
Find the ﬁrst four terms of the following recursively deﬁned sequence:
v1 = 1, v2 = 2
vn = vn−1 + vn−2 + 1, n ≥ 3
Problem 22.2
Prove each of the following for the Fibonacci sequence:
2 2
a. Fk − Fk−1 = Fk Fk+1 − Fk+1 Fk−1 , k ≥ 1.
2 2 2
b. Fk+1 − Fk − Fk−1 = 2Fk Fk−1 , k ≥ 1.
2 2
c. Fk+1 − Fk = Fk−1 Fk+2 , k ≤ 1.
d. Fn+2 Fn − Fn+1 = (−1)n for all n ≥ 0.
2
Problem 22.3
Find limn→∞ FFn where F0 , F1 , F2 , · · · is the Fibonacci sequence. (Assume
n+1
that the limit exists.)
Problem 22.4
Deﬁne x0 , x1 , x2 , · · · as follows:
xn = 2 + xn−1 , x0 = 0.
Find limn→∞ xn .
Problem 22.5
a. Make a list of all bit strings of lengths zero, one, two, three, and four that
do not contain the pattern 111.
b. For each n ≥ 0 let dn = the number of bit strings of length n that do not
contain the bit pattern 111. Find d0 , d1 , d2 , d3 , and d4 .
c. Find a recurrence relation for d0 , d1 , d2 , · · ·
d. Use the results of (b) of (c) to ﬁnd the number of bit strings of length ﬁve
that do not contain the pattern 111.
Problem 22.6
Find a formula for each of the following sums:
a. 1 + 2 + · · · + (n − 1), n ≥ 2.
b. 3 + 2 + 4 + 6 + 8 + · · · + 2n, n ≥ 1.
c. 3 · 1 + 3 · 2 + 3 · 3 + · · · 3 · n, n ≥ 1.
142.
142 RELATIONS AND FUNCTIONS
Problem 22.7
Find a formula for each of the following sums:
a. 1 + 2 + 22 + · · · + 2n−1 , n ≥ 1.
b. 3n−1 + 3n−2 + · · · + 32 + 3 + 1, n ≥ 1.
c. 2n + 3 · 2n−2 + 3 · 2n−3 + · · · + 3 · 22 + 3 · 2 + 3, n ≥ 1.
d. 2n − 2n−1 + 2n−2 − 2n−3 + · · · + (−1)n−1 · 2 + (−1)n , n ≥ 1.
Problem 22.8
Use iteration to guess a formula for the following recusively deﬁned sequence
and then use mathematical induction to prove the validity of your formula:
c1 = 1, cn = 3cn−1 + 1, for all n ≥ 2.
Problem 22.9
Use iteration to guess a formula for the following recusively deﬁned sequence
and then use mathematical induction to prove the validity of your formula:
w0 = 1, wn = 2n − wn−1 , for all n ≥ 2.
Problem 22.10
Determine whether the recursively deﬁned sequence: a1 = 0 and an = 2an−1 +
n − 1 satisﬁes the explicit formula an = (n − 1)2 , n ≥ 1.
Problem 22.11
Which of the following are second-order homogeneous recurrence relations
with constant coeﬃcients?
a. an = 2an−1 − 5an−2 .
b. bn = nbn−1 + bn−2 .
c. cn = 3cn−1 · c2 .
n−2
d. dn = 3dn−1 + dn−2 .
e. rn = rn−1 − rn−2 − 2.
f. sn = 10sn−2 .
Problem 22.12
Let a0 , a1 , a2 , · · · be the sequence deﬁned by the explicit formula
an = C · 2n + D, n ≥ 0
where C and D are real numbers.
a. Find C and D so that a0 = 1 and a1 = 3. What is a2 in this case?
b. Find C and D so that a0 = 0 and a1 = 2. What is a2 in this case?
143.
22 RECURSION 143
Problem 22.13
Let a0 , a1 , a2 , · · · be the sequence deﬁned by the explicit formula
an = C · 2n + D, n ≥ 0
where C and D are real numbers. Show that for any choice of C and D,
an = 3an−1 − 2an−2 , n ≥ 2.
Problem 22.14
Let a0 , a1 , a2 , · · · be the sequence deﬁned by the explicit formula
a0 = 1, a1 = 2
an = 2an−1 + 3an−2 , n ≥ 2
Find an explicit formula for the sequence.
Problem 22.15
Let a0 , a1 , a2 , · · · be the sequence deﬁned by the explicit formula
a0 = 1, a1 = 4
an = 2an−1 − an−2 , n ≥ 2
Find an explicit formula for the sequence.
Problem 22.16
The triangle inequality for absolute value states that for all real numbers a
and b, |a+b| ≤ |a|+|b|. Use the recursive deﬁnition of summation, the triangle
inequality, the deﬁnition of absolute value, and mathematical induction to
prove that for all positive integers n, if a1 , a2 , · · · , an are real numbers then
n n
| ak | ≤ |ak |.
k=1 k=1
Problem 22.17
Use the recursive deﬁnition of union and intersection to prove the following
general distributive law: For all positive integers n, if A and B1 , B2 , · · · , Bn
are sets then
A ∩ (∪n Bk ) = ∪n (A ∩ Bk ).
k=1 k=1
144.
144 RELATIONS AND FUNCTIONS
Problem 22.18
Use mathematical induction to prove the following generalized De Morgan’s
law.
(∩n Ai )c = ∪n Ac
i=1 i=1 i
Exercise 367
Show that the relation F : IN → Z given by the rule
1 if n = 1.
n
F (n) = F(2) if n is even
1 − F (5n − 9) if n is odd and n > 1
does not deﬁne a function.
145.
23 PROJECT VII: APPLICATIONS TO RELATIONS 145
23 Project VII: Applications to Relations
Part I: Relational Database
The ”bi” in binary relation R refers to the fact that R is a subset of the
cartesian product of two sets. Let A1 , A2 , · · · , An be given sets. If R is a
subset of A1 × A2 × · · · × An then we call R an n-ary relation. An n-ary
relation can be represented by a table or a set of ordered n-tuples.
Example 23.1
ID# Name Position Age
22012 Johnsonbaugh c 22
93831 Glover of 24
58199 Battey p 18
84341 Cage c 30
01180 Homer lb 37
26710 Score p 22
61049 Johnsonbaugh of 30
39826 Singleton 2b 31
A database is a collection of records that are manipulated by a computer.
Database management systems are programs that help users access the
information in databases. The relational database model is based on the
concept of an n-ary relation.
When an n-ary relation is represented by a table then the columns in this ta-
ble are called attributes. In the above table, the attributes are ID Number,
Name, Position, and Age. A single attribute or a combination of attributes
for a relation is called a key if the values of the attributes uniquely deﬁne
an n-tuple. For example, in the above table, we can take the attribute ID
Number as a key since every person has a unique identiﬁcation number. The
attribute Name is not a key because diﬀerent persons can have the same
name. For the same reason, we cannot take the attribute Age as a key. A
database management system responds to queries. A query is a request for
information from the database. For example, ”Find all persons that are 22
years old”.
146.
146 RELATIONS AND FUNCTIONS
Problem 23.1
a. Express the above 4-ary as a set of 4-tuples.
b. Answer the query: PLAYER[Name]
c. Answer the query: PLAYER[Name, Position]
Part II: Representing a Relation by a Matrix.
Let A be a set with n elements and R be a binary relation on A. Deﬁne
the n × n matrix M (R) = (mij ) as follows:
1 if (ai , bj ) ∈ R
mij =
0 if (ai , bj ) ∈ R
If the numbers on the main diagonal of M (R) are all equal to 1 then R is
reﬂexive. If M (R)T = M (R), where M (R)T is the transpose of M (R), then
the relation R is symmetric. If mij = 0 or mji = 0 for i = j then R is
antisymmetric.
Problem 23.2
Let A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.
Find M (R) and use it to determine if the relation R is reﬂexive, symmetric
or antisymmetric.
Part III: Cryptology
An important application to congruences is cryptology, which is the study of
secret messages.
(a) The process of making a message secret is called encryption. This
process consists of assigning the alphabet A, B, C, · · · , Z by the integers
0, 1, 2, · · · , 25. Then the encrypted version of the message, the letter rep-
resented by p is replaced with the letter represented by the remainder of the
division of (p + 3) by 26.
Problem 23.3
What is the encrypted message produced from the message ”MEET YOU
IN THE PARK”?
(b) Decryption is the process of determining the original message. In this
case the letter represented by p is replaced by the letter represented by the
remainder of the division of (p − 3) by 26.
147.
23 PROJECT VII: APPLICATIONS TO RELATIONS 147
Problem 23.4
What is the message produced from the encrypted message ”PHHW BRX
LQ WKH SDUN”?
148.
148 RELATIONS AND FUNCTIONS
24 Project VIII: Well-Ordered Sets and Lat-
tices
Let [A, ≤] be a poset. Let B ⊆ A. An element b ∈ B is called a least
element of B if and only if b ≤ x for all x ∈ B. If x ≥ b for all x ∈ B then
we call b the greatest element of B.
Problem 24.1
Consider the set IN with the inequality relation ≤ . Let B = {2, 4, 5, 6, 7, 8, 9}.
What is the least element of B? What is the greatest element of B?
A poset [A, ≤] is said to be well-ordered if and only if ≤ is a total order
and every subset of A has a least element.
Problem 24.2
a. Show that (IN, ≤) is well-ordered.
b. Show that (Z , ≤) is not well-ordered.
Z
An element a ∈ A is called a lower bound of B if a ≤ x for all x ∈ B. We
call a ∈ A an upper bound of B if x ≤ a for all x ∈ B. Note that a lower
bound or an upper bound is not unique.
Problem 24.3
Consider the poset [IN, ≤]. Let B = {2, 4, 8, 10}. Find a lower bound of B as
well as an upper bound.
The greatest element of the set of lower bounds of B is called the greatest
lower bound, in symbol g.l.b(B). The least upper bound of the set of upper
bounds of B is called the least upper bound, in symbol l.u.b(B).
Problem 24.4
Consider the poset [IR, ≤] and B = (−1, 1). Find g.l.b(B) and l.u.b(B).
A lattice is a poset [A, ≤] such that every pair of elements in A have a l.u.b
and g.l.b in A.
Problem 24.5
Show that [IR, ≤] is a lattice.
Problem 24.6
Let A = {2, 3, 4, 9, 12, 18} and R be the binary relation ”divides” on A. Show
that [A, R] is not a lattice.
149.
25 PROJECT IX: THE PIGEONHOLE PRINCIPLE 149
25 Project IX: The Pigeonhole Principle
The Pigeonhole principle asserts that if n pigeons ﬂy into k holes with
n > k then some of the pigeonholes contain at least two pigeons. The
reason this statement is true can be seen by arguing by contradiction. If the
conclusion is false, each pigeonhole contains at most one pigeon and, in this
case, we can account for at most k pigeons. Since there are more pigeons
than holes, we have a contradiction.
Problem 25.1
Ten persons have ﬁrst names George, William, and Laura and last names
Bush, Perry, and Gramm. Show that at least two persons have the same
ﬁrst and last names.
A mathematical way to formulate the pigeonhole principle is given by the
following exercise
Problem 25.2
Let S be a ﬁnite set and {A1 , A2 , · · · , An } be a partition of S. Use the method
of contradiction to show that there is an index 1 ≤ i ≤ n such that |Ai | ≥ |S| . n
One can uses the previous exercise to solve the following exercise.
Problem 25.3
Let S and T be two ﬁnite sets such that |S| > k|T | where k is a positive
integer. Show that for any function f : S → T there is a t ∈ T such that the
set {s ∈ S : f (s) = t} has more than k elements.
Hint: Show that the family At = {s ∈ S : f (s) = t}, where t ∈ T, partitions
S into n sets with n ≤ |T |. Then apply the previous exercise.
As a consequence of the above exercise we have
Problem 25.4
If S and T are ﬁnite sets such that |S| > |T | then any function f : S → T is
not one-to-one.
150.
150 RELATIONS AND FUNCTIONS
26 Project X: Countable Sets
We say that two sets have the same cardinality if and only if there is a
bijective function between them. A set A is called countably inﬁnite if
and only if A has the same cardinality as the set IN∗ of positive integers. A
set A is called countable if it is either ﬁnite or countably inﬁnite. A set
that is not countable is said to be uncountable. Examples of uncountable
sets are IR and the intervals in IR..
The purpose of this project is to look at some countably inﬁnite sets.
Problem 26.1
Show that the function f : IN∗ → IN given by f (n) = n − 1 is a bijective
function. Thus, IN is countably inﬁnite.
Problem 26.2
Show that the function f : IN∗ → Z deﬁned by
n
2
if n is even
f (n) = 1−n
2
if n is odd
is bijective. Hence, Z is countably inﬁnite.
Problem 26.3
Show that the function f : Z → 2Z deﬁned by f (n) = 2n is a bijective
Z
function. Hence, the set of even integers is countably inﬁnite.
Problem 26.4
Show that the set of rational numbers Q is countably inﬁnite. (Hint:
I
152.
152 RELATIONS AND FUNCTIONS
27 Project XI: Finite-State Automaton
A ﬁnite-state machine can be looked at as a mathematical model that can
accept input, store and process information and produce output. Examples
include digital computers, compilers, vending machines, coin changers, tele-
phones, and elevators.
This model has an input/output unit, and, consequently, has a way of com-
municating with the world using a set of symbols. Let I be the set of input
symbols and O, the set of output symbols. In the case of an elevator I might
be up, down, and ﬂoor selection, while O might be stops on particular ﬂoors.
Besides input and output symbols, there is a set of states S for our model. A
state is like a snapshot of what is happening in the machine at a particular
instant. An elevator might be in a state of going down to the ﬁrst ﬂoor to
pick up a passenger or in a state of stopping on the third ﬂoor on the way
up to the ﬁfth ﬂoor. There are always an intial state of our model, denoted
by s0 , and ﬁnal or accepting state(s).
Also, our model has a function, called the next state function. This function
returns the next state based on the present state and input. For instance, if
the elevator is in the state of moving up to the ﬁfth ﬂoor and has an input
of someone pressing the down button on the third ﬂoor, it goes to a state
that says,” Remember, when coming back down from stopping and picking
someone on the third ﬂoor.”
The above discussion is formalized as follows:
A ﬁnite-state automaton A consists of ﬁve objects:
1. a set I, called the input alphabet, of input symbols;
2. a set O, called the output alphabet, of output symbols;
3. a set S of states the machine can be in;
4. a subset of S whose elements are called accepting states;
5. a next-state function or transition function N : S × I → S. If s ∈ S
and m ∈ I then N (s, m) is the state to which A goes if m is input to A when
A is in state s. The initial state of the machine is s0 .
The operation of a ﬁnite-state machine is commomly described by a dia-
gram called a transition diagram. The edges labeled with inputs and
nodes with states. A double circle stands for the ﬁnal or accepting state(s).
Problem 27.1
Consider the ﬁnite-state automaton deﬁned by the transition diagram
153.
27 PROJECT XI: FINITE-STATE AUTOMATON 153
a. What are the elements of S?
b. What are the input symbols?
c. What is the initial state?
d. What are the accepting states?
e. Find N (s3 , 1) and N (s3 , 0).
Let A be a ﬁnite-state automaton with input alphabet I. Let I ∗ be the set
of all words with letters from I. A word w ∈ I ∗ is said to be accepted by
A if, and only if, A goes to an accepting state when the symbols of w are
input to A in sequence starting when A is in its initial state. The language
accepted by A, denoted by L(A), is the set of all words that are accepted
by A.
Problem 27.2
Consider the ﬁnite-state automaton deﬁned by the following transition dia-
gram
154.
154 RELATIONS AND FUNCTIONS
a. To what states does A go if the symbols of the following words are input
to A in sequence starting from the initial state?
(i) 1101 (ii) 0011 (iii) 0101010.
b. Which of the words in part (a) send A to an accepting state?
c. Show that L(A) = {0(10)n : n ≥ 0} where (10)n = 1010 · · · with n copies
of 10 juxtaposed into one word.
Let A be a ﬁnite-state automaton with input alphabet I and states S. Let
N ∗ : S × I ∗ → S be the function deﬁned as follows: N ∗ (s, w) is the state to
which A goes if the symbols of w are input to A in sequence starting when
A is in state s. We call N ∗ the eventual -state function.
Problem 27.3
A ﬁnite-state automaton A, given by the transition diagram below, has tran-
sition function N and eventual-state functioin N ∗ .
155.
27 PROJECT XI: FINITE-STATE AUTOMATON 155
a. Find N (s2 , 0) and N (s1 , 0).
b. Find N ∗ (s2 , 11010) and N ∗ (s0 , 01000).
Problem 27.4
Design a ﬁnite-state machine that recognizes words of the form 01, 011, 0111, 01111, · · · .
157.
Introduction to the Analysis of
Algorithms
Informally, an algorithm is any well-deﬁned computational procedure that
takes a set of values as input and produces a set of values as output.
The subject of the analysis of algorithms consists of the study of eﬃciency of
algorithms. Two aspects of the algorithm eﬃciency are: the amount of time
required to execute the algorithm and the memory space it consumes. In this
chapter we introduce the basic techniques for calculating time eﬃciency.
28 Time Complexity and O-Notation
The primary eﬃciency criterion for analyzing the eﬃciency of an algorithm
is the running time of the algorithm as a function of the number of values
it processes. The running time of an algorithm is not measured by counting
the minutes and seconds for the algorithm written in a particular language
and running on a particular machine. Rather it is deﬁned to be an estimate
of the number of operations performed by the algorithm given a particular
number of input values.
Generally, given an algorithm that performs a task, we will be interested in
estimating the running time as a function of the problem size. For example,
let us consider the sequential search of an item X from a list of n items.
Here, we say that the problem size is n. Let T (n) be a measure of the time
required to execute an algorithm of problem size n. We call T (n) the time
complexity function of the algorithm. If n is suﬃciently small then the
algorithm will not have a long running time. Thus, the interesting question
is:”How fast T (n) increases as n increases?” This is called the asymptotic
behavior of the time complexity function.
In our time analysis we will restrict ourselves to the worst case behavior of
157
158.
158 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
an algorithm; that is, the longest running time for any input of size n.
Since we are considering asymptotic eﬃciency of algorithms, basically we will
be focusing on the leading term of T (n). For example, if T (n) = 4n3 − 2n2 +
2 1 5
n + 5 then T (n) = n3 (4 − n + n + n ) and for large n we have T (n) ∼ n3 . We
say that T (n) has a growth of order n3 .
We say that one algorithm is more eﬃcient then another if its worse case
running time has a lower order of growth.
Example 28.1
Estimate the time complexity of the following algorithm:
i := 1
p := 1
for i := 1 to n
p := p · i
i := i + 1
next i
Solution.
Prior to entering the loop, it takes two assignment statements to initialize
the variables i and p. The loop is executed n times, and each time it executes
the two assignment statements in the body of the loop with a total of two
arithmetic operations. Thus, the time complexity of the algorithm is given
by
T (n) = 4n + 2
so the growth is of order n.
Example 28.2
What is the run-time complexity based on n for the following program sege-
ment:
for i := 1 To n
for j := 1 To n
A(i,j) := x
next j
next i
Solution.
The inner loop is executed n times and the outer loop also is executed n
159.
28 TIME COMPLEXITY AND O-NOTATION 159
times. Hence, T (n) = n2 so that the growth is of order n2 .
In the above two problems we found a precise expression for the time com-
plexity of the algorithm. What usually interests us is the order of growth.
We next introduce some of the concepts of growth orders. Let g : IN → IR.
We deﬁne the set
O(g(n)) = {f (n) : there exists positive constants n0 and C such that
|f (n)| ≤ C|g(n)|, f or n ≥ n0 }.
We say that a function f is order at most g or f big-oh of g if and
only if f (n) ∈ O(g(n)). Sometimes We write f (n) = O(g(n)). Graphically,
this means that for n ≥ n0 the graph of f is below the graph of g.
Example 28.3
Show that the time complexity found in Exercise 390 is O(n).
Solution.
To show that T (n) = O(n) we must produce constants C and n0 such that
T (n) ≤ Cn for n ≥ n0 . Indeed, T (n) ≤ 5n for n ≥ 2 so that n0 = 2 and
C = 5.
Example 28.4
Show that the run-time complexity based on n for the following program
segement is O(n2 ).
s := 0
for i := 1 To n
for j := 1 To i
s := s + j · (i − j + 1)
next j
next i
Solution.
Prior to entering the loop there is one assignment statement. Now, there are
two additions, one subtraction, one multiplication and one assignment for
each iteration of the inner loop. The total number of time the inner loop is
executed is
n(n + 1)
1 + 2 + 3 + ··· + n =
2
160.
160 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
n(n+1)
Hence, T (n) = 5 · 2
+ 1 ≤ 5n2 , n ≥ 1 so that C = 5 and n0 = 1. Hence,
T (n) ∈ O(n2 ).
We say that a function f is of polynomial complexity if and only if
f ∈ O(np ) for some p ∈ IN. If p = 0 then we say that f is of constant
complexity. If p = 1 we say that the complexity is linear.
Example 28.5
Show that
1 + 2 + 3 + · · · + n = O(n2 ).
Solution.
Indeed, since n ≥ 1 then
1 + 2 + 3 + · · · + n ≤ n + n + n + · · · + n = n2
so that C = 1 and n0 = 1.
Example 28.6
Show that n3 ∈ O(n2 ).
Solution.
We proceed by contradiction. Suppose that n3 ∈ O(n2 ). Then there exist
constants C and n0 such that n3 ≤ Cn for all n ≥ n0 . Dividing through by
n2 to obtain n ≤ C. This leads to a contradiction since the left-hand side can
be made as large as we please whereas the right-hand side is constant.
Example 28.7
Show that if f (n) ∈ O(g(n)) and g(n) ∈ O(h(n)) then f (n) ∈ O(h(n)).
Solution.
Since f (n) ∈ O(g(n)), there exist n1 and C1 such that |f (n)| ≤ C1 |g(n)| for all
n ≥ n1 . Similarly, there exist constants C2 and n2 such that |g(n)| ≤ C2 |h(n)|
for all n ≥ n2 . Let n0 = max{n1 , n2 } and C = C1 C2 . Then for n ≥ n0 we
have
|f (n)| ≤ C1 |g(n)| ≤ C1 C2 |h(n)| = C|h(n)|
161.
28 TIME COMPLEXITY AND O-NOTATION 161
Example 28.8
Suppose we want to arrange the elements of a one dimensional array a[1], a[2], · · · , a[n]
in increasing order. An insertion sort compares every pair of elements,
switching the values of those that are out of order, a[i − 1] > a[i].
a. How many possible pairs are compared?
b. What is the maximum number of exchanges?
c. What time the complexity of this algorithm in the worst case?
d. Is this a polynomial-time algorithm?
Solution.
a. The number of possible pairs to compare in the algorithm is
n(n − 1)
1 + 2 + · · · + (n − 1) = .
2
n(n−1)
b. From part a. it follows that the maximum number of exchanges is 2
.
c. T (n) = n(n−1) .
2
2
d. For n ≥ 1, T (n) ≤ n so that T (n) ∈ O(n2 ).
2
Next, we recall the following deﬁnition from calculus. If L = limx→∞ f (x)
then for any > 0 there is a positive integer N such that |f (x) − L| <
whenever n ≥ N.
Using this deﬁnition, we have the following important theorem.
Theorem 28.1
Suppose that limn→∞ f (n) = L with L ≥ 0. Then f (n) ∈ O(g(n)). Moreover,
g(n)
a. if L > 0 then g(n) ∈ O(f (n)), and
b. if L = 0 then g(n) ∈ O(f (n)).
Proof.
Let = 1. Then there is a positive integer n0 such that | f (n) −L| < 1 whenever
g(n)
n ≥ n0 . This implies that | f (n) | < 1 + L for n ≥ n0 . Hence, |f (n)| < C|g(n)|
g(n)
where C = (1 + L) and n ≥ n0 . But this is just saying that f (n) ∈ O(g(n)).
g(n) 1
a. Now, suppose that L > 0. Then limn→∞ f (n) = L . Interchange the roles
of f and g in the previous argument to ﬁnd that |g(n)| < C|f (n)| where
1
C = 1 + L and n ≥ n0 for some positive integer n0 . Hence, g(n) ∈ O(f (n)).
b. Now suppose that L = 0. We use contradiction to show that g(n) ∈
O(f (n)). So suppose that g(n) ∈ O(f (n)). Then there exist positive constants
162.
162 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
C and M1 such that |g(n)| ≤ C|f (n)| for all n ≥ M1 . On the other hand, by
letting = C we can ﬁnd a positive integer M2 such that | f (n) | < whenever
1
g(n)
n ≥ M2 . Let n0 = max{M1 , M2 }. Then for n ≥ n0 we have
g(n)
C<| |≤C
f (n)
which is a contradiction. Hence, we must have g(n) ∈ O(f (n)).
163.
28 TIME COMPLEXITY AND O-NOTATION 163
Review Problems
Problem 28.1
Find the worst case running time of the following segment of an algorithm:
for i := 1 to n
for j := 1 to i+1 2
a := (n − i) · (n − j)
next j
next i
Problem 28.2
Find the worst case running time of the following segment of an algorithm:
for i := 1 to n
for j := 1 to 2n
for k := 1 to n
x := i · j · k
next k
next j
next i
Problem 28.3
Construct a table showing the result of each step when insertion sort is
applied to the array a[1] = 6, a[2] = 2, a[3] = 1, a[4] = 8, a[5] = 4.
Problem 28.4
How many comparisons actually occur when insertion sort is applied to the
array of the previous exercise?
Problem 28.5
Selection sort is another algorithm for arranging the elements of a one-
dimensional array a[1], a[2], · · · , a[n] in increasing order. The sorting works
by selecting the smallest item in the list, moving it to the front of the list,
and then ﬁnding the smallest of the remaining items and moving it to the
second position in the list, and so on. When two items in the list, say a[k]
and a[m], have to be interchanged, we write switch(a[k], a[m]). The following
is the selection algorithm:
for i := 1 to n − 1
164.
164 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
min := i
for j := i + 1 to n
if a[min] > a[j] then
switch(a[min], a[i])
next j
next i
Construct a table showing the result of each step when selection sort is ap-
plied to the array a[1] = 5, a[2] = 3, a[3] = 4, a[4] = 6, a[5] = 2.
Problem 28.6
How many comparisons actually occur when selection sort is applied to the
array of the previous exercise?
Problem 28.7
√ √
Show that n ∈ O( n).
Problem 28.8
Show that
12 + 22 + · · · + n2 ∈ O(n3 ).
Problem 28.9
Show that
13 + 23 + · · · + n3 ∈ O(n4 ).
Problem 28.10
a. Use mathematical induction to show that
1 1 1 4
13 + 23 + · · · + n3 ≤ n3
for all n ≥ 1.
b. What can you conclude from part (a) about the order of the above sum?
165.
29 LOGARITHMIC AND EXPONENTIAL COMPLEXITIES 165
29 Logarithmic and Exponential Complexities
In this section we assume that the reader is familiar with the deﬁnitions and
rules of both exponential and logarithmic functions. Unless explicitly stated,
all logarithms in this chapter are to base 2 mainly because of the following
theorem
Theorem 29.1
For any a > 1, O(loga n) = O(log2 n).
Proof.
We must show that there exist constants C1 , C2 and n0 such that loga n ≤
C1 log2 n and log2 n ≤ C2 loga n for all n ≥ n0 . By the change of bases formula
we have
log2 n
loga n = .
log2 a
1
Now, let C1 = log2 a
, C2 = log2 a, and n0 = 1.
If f (n) ∈ O(log2 n) we say that f (n) has logarithmic complexity. A function
f (n) is said to be of exponential complexity if and only if f (n) ∈ O(an )
for some a > 1.
Example 29.1
Show that n + n log2 n ∈ O(n log2 n).
Solution.
n
Since limn→∞ n log n = 0, there is a positive integer n0 such that n < n log2 n
2
for all n ≥ n0 . Thus,n + n log2 n < 2nlog2 n = Cn log2 n, n ≥ n0 . This shows
that n + n log2 n ∈ O(n log2 n).
Example 29.2
a. Show that n! = O(nn ).
b. Show that n = O(2n ).
c. Use b. to show that log2 n = O(n).
Solution.
a. Since n − i ≤ n for 0 ≤ i ≤ n we have
n! = n(n − 1)(n − 2) · · · 2 · 1
≤ n · n · n · · · n · n = nn
166.
166 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
It follows that n! = O(nn ).
b. We show by induction on n ≥ 0 that n ≤ 2n .
Basis of induction: For n = 0 we have 0 ≤ 20 .
Induction hypothesis: Suppose that n ≤ 2n .
Induction step: We must show that n + 1 ≤ 2n+1 . Indeed,
n+1 ≤n+n
≤ 2n + 2n = 2n+1
Hence, n = O(2n ).
c. Take the logarithm of both sides of b. to obtain log2 n ≤ n, n ≥ 1. That
is, log2 n = O(n).
Example 29.3
a. Show that log2 n! = O(n log2 n).
b. Show that n log2 n = O(log2 n!).
Solution.
a. We have shown that n! = O(nn ). That is, n! ≤ nn for n ≥ 1. Take loga-
rithm of both sides to obtain log2 n! ≤ n log2 n. That is, log2 n! = O(n log2 n).
b. It is easy to see that (n − i)(i + 1) ≥ n for all 0 ≤ i ≤ n − 1. In this case
(n!)2 = [n · (n − 1) · · · 2 · 1][1 · 2 · · · (n − 1) · n]
= (n · 1)[(n − 1) · 2] · · · [2 · (n − 1)](1 · n)
≥ n · n···n · n
= nn .
Now take the logarithm of both sides to obtain n log2 n ≤ 2 log2 n!. That is,
n log2 n = O(log2 n!).
Example 29.4
a. Show that if f1 (n) ∈ O(g(n)) and f2 (n) ∈ O(g(n)) then f1 (n) + f2 (n) ∈
O(g(n)).
b. Show that if f1 (n) ∈ O(g1 (n)) and f2 (n) ∈ O(g2 (n)) then f1 (n) · f2 (n) ∈
O(g1 (n) · g2 (n)).
c. Use a. and b. to show that
3n log2 n! + (n2 + 3) log2 n = O(n2 log2 n).
167.
29 LOGARITHMIC AND EXPONENTIAL COMPLEXITIES 167
Solution.
a. Since f1 (n) ∈ O(g(n)) then there exist n1 and C1 such that |f1 (n)| ≤
C1 |g(n)| for all n ≥ n1 . Similarly, there exist constants C2 and n2 such that
|f2 (n)| ≤ C2 |g(n)| for all n ≥ n2 . Let n0 = max{n1 , n2 } and C = C1 + C2 .
Then for n ≥ n0 we have
|f1 (n) + f2 (n)| ≤ C1 |g(n)| + C2 |g(n)| = C|g(n)|.
b. Now since f1 (n) ∈ O(g1 (n)), there exist n1 and C1 such that |f1 (n)| ≤
C1 |g1 (n)| for all n ≥ n1 . Similarly, there exist constants C2 and n2 such that
|f2 (n)| ≤ C2 |g2 (n)| for all n ≥ n2 . Let n0 = max{n1 , n2 } and C = C1 · C2 .
Then for n ≥ n0 we have
|f1 (n) · f2 (n)| ≤ C|g1 (n)g2 (n)|.
c. Using b. above and a. of the previous exercise we have 3n log2 n! =
O(n2 log2 n). Since (n2 + 3) log2 n = O(n2 log2 n) then by a. and b. the result
follows.
168.
168 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
Review Problems
Problem 29.1
Show that 1 + 2 + 22 + · · · + 2n ∈ O(2n+1 ).
Problem 29.2
Show that 2n +
3
2n
32
+ 2n
33
+ ··· + 2n
3n
∈ O(n).
Problem 29.3
Show that n2 + 2n ∈ O(2n ).
Problem 29.4
a. Show that 2 + 1 + · · · + n ≤ ln n, n ≥ 2.
1
3
1
b. Use part a. to show that for n ≥ 3
1 1
1+ + · · · + ≤ ln n.
2 n
n n n
c. Use b. to show that n + 2
+ 3
+ ··· + n
∈ O(n ln n).
Problem 29.5
Show that 2n ∈ O(n!).
169.
30 Θ- AND Ω-NOTATIONS 169
30 Θ- and Ω-Notations
The O-notation asymptotically bounds a function from above. When we
have bounds from above and below, we use Θ notation. For a given function
g(n), we denote by Θ(g(n)) to be the set of all functions f such that there
exist positive constants C1 , C2 , and n0 such that C1 |g(n)| ≤ |f (n)| ≤ C2 |g(n)|
for all n ≥ n0 . If f ∈ Θ(g(n)) we write f (n) = Θ(g(n)).
Example 30.1
1
Show that 2 n2 − 3n = Θ(n2 ).
Solution.
Let C1 and C2 be positive constants such that
1
C1 n2 ≤ n2 − 3n ≤ C2 n2 .
2
This is equivalent to
1 3
C1 ≤ − ≤ C2 .
2 n
1
Since 2 − n ≤ 1 for all n ≥ 1, we choose C2 ≥ 1 . Since
3
2 2
1
2
− 3
n
≥ 1
4
for all
n ≥ 12, we choose C1 ≤ 1 . Finally, we choose n0 = 12.
4
Example 30.2
Show that 6n3 = Θ(n2 ).
Solution.
We use the argument by contradiction. Suppose that 6n3 = Θ(n2 ). Then
there exist positive constants C1 , C2 and n0 such that
C1 n2 ≤ 6n3 ≤ C2 n2
for all n ≥ n0 . The right-hand side inequality yields 6n ≤ C2 for n ≥ n0 .
This says that the left-hand side can be made as large as we want whereas
the right-hand side is ﬁxed. A contradiction.
Theorem 30.1
For given two functions f (n) and g(n), f (n) = Θ(g(n)) if and only if f (n) =
O(g(n)) and g(n) = O(f (n)).
170.
170 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
Proof.
Suppose that f (n) = Θ(g(n)). Then there exist positive constants C1 , C2 ,
and n0 such that C1 |g(n)| ≤ |f (n)| ≤ C2 |g(n)| for all n ≥ n0 . The left-hand
side inequality implies that g(n) = O(f (n)) whereas the right-hand side in-
equality implies that f (n) = O(g(n)). Now go backward for the converse.
Just as O provides an asymptotic upper bound on a function, Ω−notation
provides an asymptotic lower bound. For a given function g(n), let Ω(g(n))
denote the set of all funtions f (n) such that there exist positive constants
C and n0 such that C|g(n)| ≤ |f (n)| for all n ≥ n0 . For f (n) ∈ Ω(g(n)) we
write f (n) = Ω(g(n)).
Example 30.3
Show that log2 n! = Ω(n log2 n).
Solution.
1
Since (n!)2 ≥ nn for all n ≥ 1 we ﬁnd n log2 n ≤ 2 log2 n!. That is, 2 n log2 n ≤
log2 n! for n ≥ 1. This says that log2 n! = Ω(n log2 n).
Theorem 30.2
For given two functions f (n) and g(n), f (n) = Θ(g(n)) if and only if f (n) =
O(g(n)) and f (n) = Ω(g(n)).
Proof.
Suppose ﬁrst that f (n) = Θ(g(n)). Then there exist positive constants C1 , C2
and n0 such that C1 |g(n)| ≤ |f (n)| ≤ C2 |g(n)| for n ≥ n0 . The right-hand
side inequality implies that f (n) = O(g(n)) whereas the left-hand side in-
equality implies that f (n) = Ω(g(n)).
Conversely, suppose that f (n) = O(g(n)) and f (n) = Ω(g(n)). Then there
exist constants C1 , C2 , n1 and n2 such that |f (n)| ≤ C2 |g(n)| for n ≥ n2 and
C1 |g(n)| ≤ |f (n)| for n ≥ n1 . Let n0 = max{n1 , n2 }. Then for n ≥ n0 we
have C1 |g(n)| ≤ |f (n)| ≤ C2 |g(n)|. That is, f (n) = Θ(g(n)).
Example 30.4
Let f (n) and g(n) be two given functions. We say that f (n) = o(g(n)) if and
only if limn→∞ f (n) = 0.
g(n)
a. Show that if f (n) = o(g(n)) then f (n) = O(g(n)).
b. Find two functions f (n) and g(n) such that f (n) = O(g(n) but f (n) =
o(g(n)).
171.
30 Θ- AND Ω-NOTATIONS 171
Solution.
a. Suppose that f (n) = o(g(n)). Then there is a positive integer n0 such
that | f (n) | ≤ 1 for n ≥ n0 . That is, |f (n)| ≤ |g(n)| for all n ≥ n0 . Hence,
g(n)
f (n) = O(g(n)).
b. Let f (n) = 2n2 and g(n) = n2 .
172.
172 INTRODUCTION TO THE ANALYSIS OF ALGORITHMS
173.
Fundamentals of Counting and
Probability Theory
The major goal of this chapter is to establish several techniques for counting
large ﬁnite sets without actually listing their elements. Also, the fundamen-
tals of probablity theory are discussed.
31 Elements of Counting
For a set X, |X| denotes the number of elements of X. It is easy to see that
for any two sets A and B we have the following result known as the Inclu-
sion - Exclusion Principle
|A ∪ B| = |A| + |B| − |A ∩ B|.
Indeed, |A| gives the number of elements in A including those that are com-
mon to A and B. The same holds for |B|. Hence, |A| + |B| includes twice
the number of common elements. Hence, to get an accurate count of the
elements of A ∪ B, it is necessary to subtract |A ∩ B| from |A| + |B|.
Note that if A and B are disjoint then |A ∩ B| = 0 and consequently
|A ∪ B| = |A| + |B|.
Example 31.1 (The Addition Rule)
Show by induction on n, that if {A1 , A2 , · · · , An } is a collection of pairwise
disjoint sets then
|A1 ∪ A2 ∪ · · · ∪ An | = |A1 | + |A2 | + · · · + |An |.
Solution.
Basis of induction: For n = 2 the result holds by the Inclusion-Exclusion
173
174.
174 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Principle.
Induction hypothesis: Suppose that for any collection {A1 , A2 , · · · , An } of
pairwise disjoint sets we have
|A1 ∪ A2 ∪ · · · ∪ An | = |A1 | + |A2 | + · · · + |An |.
Induction step: Let {A1 , A2 , · · · , An , An+1 } be a collection of pairwise disjoint
sets. Since (A1 ∪ A2 ∪ · · · ∪ An ) ∩ An+1 = (A1 ∩ An+1 ) ∪ · · · ∪ (An ∩ An+1 ) = ∅
then by the Inclusion-Exclusion Principle and the induction hypothesis we
have
|A1 ∪ A2 ∪ · · · ∪ An ∪ An+1 | = |A1 ∪ A2 ∪ · · · ∪ An | + |An+1 |
= |A1 | + |A2 | + · · · + |An | + |An |
Example 31.2
A total of 35 programmers interviewed for a job; 25 knew FORTRAN, 28
knew PASCAL, and 2 knew neither languages. How many knew both lan-
guages?
Solution.
Let A be the group of programmers that knew FORTRAN, B those who
knew PASCAL. Then A ∩ B is the group of programmers who knew both
languages. By the Inclusion-Exclusion Principle we have
|A ∪ B| = |A| + |B| − |A ∩ B|.
That is,
33 = 25 + 28 − |A ∩ B|.
Solving for |A ∩ B| we ﬁnd |A ∩ B| = 20.
Another important rule of counting is the multiplication rule. It states
that if a decision consists of k steps, where the ﬁrst step can be made in n1
diﬀerent ways, the second step in n2 ways, · · · , the kth step in nk ways, then
the decision itself can be made in n1 n2 · · · nk ways.
Example 31.3
a. How many possible outcomes are there if 2 distinguishable dice are rolled?
b. Suppose that a state’s license plates consist of 3 letters followed by four
digits. How many diﬀerent plates can be manufactured? (no repetitions)
175.
31 ELEMENTS OF COUNTING 175
Solution.
a. By the multiplication rule there are 6 × 6 = 36 possible outcomes.
b. By the multiplication rule there are 26 × 25 × 24 × 10 × 9 × 8 × 7 possible
license plates.
Example 31.4
Let Σ = {a, b, c, d} be an alphabet with 4 letters. Let Σ2 be the set of all
words of length 2 with letters from Σ. Find the number of all words of length
2 where the letters are not repeated. First use the product rule. List the
words by means of a tree diagram.
Solution.
By the multiplication rule there are 4 × 3 = 12 diﬀerent words. Constructing
a tree diagram
we ﬁnd that the words are
{ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc}
An r-permutation of n objects, in symbol P (n, r), is an ordered selection
of r objects from a given n objects.
Example 31.5
n!
a. Use the product rule to show that P (n, r) = (n−r)! .
b. Find all possible 2-permutations of the set {1, 2, 3}.
Solution.
a. We can treat a permutation as a decision with r steps. The ﬁrst step
can be made in n diﬀerent ways, the second in n − 1 diﬀerent ways, ...,
the rth in n − r + 1 diﬀerent ways. Thus, by the multiplication rule there
are n(n − 1) · · · (n − r + 1) r-permutations of n objects. That is, P (n, r) =
n!
n(n − 1) · · · (n − r + 1) = (n−r)! .
3!
b. P (3, 2) = (3−2)! = 6.
176.
176 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Example 31.6
How many license plates are there that start with three letters followed by 4
digits (no repetitions)?
Solution.
P (26, 3) · P (10, 4) = 78, 624, 000.
An r-combination of n objects, in symbol C(n, r), is an unordered selec-
tion of r of the n objects. Thus, C(n, r) is the number of ways of choosing
r objects from n given objects without taking order in account. But the
number of diﬀerent ways that r objects can be ordered is r!. Since there are
C(n, r) groups of r objects from a given n objects then the number of ordered
selection of r objects from n given objects is r!C(n, r) = P (n, r). Thus
P (n, r) n!
C(n, r) = = .
r! r!(n − r)!
Example 31.7
In how many diﬀerent ways can a hand of 5 cards be selected from a deck of
52 cards?(no repetition)
Solution.
C(52, 5) = 2, 598, 960.
Example 31.8
Prove the following identities:
a. C(n, 0) = C(n, n) = 1 and C(n, 1) = C(n, n − 1) = n.
b. Symmetry property: C(n, r) = C(n, n − r), r ≤ n.
c. Pascal’s identity: C(n + 1, k) = C(n, k − 1) + C(n, k), n ≥ k.
Solution.
a. Follows immediately from the deﬁnition of of C(n, r).
b. Indeed, we have
n!
C(n, n − r) = (n−r)!(n−n+r)!
n!
= r!(n−r)!
= C(n, r)
177.
31 ELEMENTS OF COUNTING 177
c.
n! n!
C(n, k − 1) + C(n, k) = (k−1)!(n−k+1)!
+ k!(n−k)!
= n!k
k!(n−k+1)!
+ n!(n−k+1)
k!(n−k)!
n!
= k!(n−k+1)!
(k + n − k + 1)
(n+1)!
= (n+1−k)!
= C(n + 1, k)
Pascal’s identity allows one to construct the following triangle known as
Pascal’s triangle (for n = 5) as follows
1
1 → 1
1 → 2 → 1
1 → 3 → 3 →1
1 → 4 → 6 →4 → 1
The following theorem provides an expension of (x + y)n where n is a non-
negative integer.
Theorem 31.1 (Binomial Theorem)
Let x and y be variables, and let n be a positive integer. Then
n
n
(x + y) = C(n, k)xn−k y k
k=0
where C(n, k) is called the binomial coeﬃcient.
Proof.
The proof is by induction.
Basis of induction: For n = 1 we have
1
1
(x + y) = C(1, k)x1−k y k = x + y.
k=0
Induction hypothesis: Suppose that the theorem is true for n.
Induction step: Let us show that it is still true for n + 1. That is
n+1
n+1
(x + y) = C(n + 1, k)xn−k+1 y k .
k=0
178.
178 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Indeed, we have
(x + y)n+1 = (x + y)(x + y)n = x(x + y)n + y(x + y)n
n n
n−k k
= x C(n, k)x y +y C(n, k)xn−k y k
k=0 k=0
n n
= C(n, k)xn−k+1 y k + C(n, k)xn−k y k+1
k=0 k=0
n+1
= C(n, 0)x + C(n, 1)x y + C(n, 2)xn−1 y 2
n
+ · · · + C(n, n)xy n + C(n, 0)xn y
+ C(n, 1)xn−1 y 2 + · · · + C(n, n − 1)xy n
+ C(n, n)y n+1
= C(n + 1, 0)xn+1 + C(n + 1, 1)xn y + C(n + 1, 2)xn−1 y 2
+ · · · + C(n + 1, n)xy n + C(n + 1, n + 1)y n+1
n+1
= C(n + 1, k)xn−k+1 y k .
k=0
Example 31.9
Expand (x + y)6 using the binomial theorem.
Solution.
By the Binomial Theorem and Pascal’s triangle we have
(x + y)6 = x6 + 6x5 y + 15x4 y 2 + 20x3 y 3 + 15x2 y 4 + 6xy 5 + y 6
Example 31.10
a. Show that n C(n, k) = 2n .
k=0
b. Show that n (−1)k C(n, k) = 0.
k=0
Solution.
a. Letting x = y = 1 in the binomial theorem we ﬁnd
n
n n
2 = (1 + 1) = C(n, k).
k=0
b. This follows from the binomial theorem by letting x = 1 and y = −1
179.
31 ELEMENTS OF COUNTING 179
Review Problems
Problem 31.1
a. How many ways can we get a sum of 4 or a sum of 8 when two distin-
guishable dice are rolled?
b. How many ways can we get a sum of 8 when two undistinguishable dice
are rolled?
Problem 31.2
a. How many 4-digit numbers can be formed using the digits, 1, 2, · · · , 9
(with repetitions)? How many can be formed if no digit can be repeated?
b. How many diﬀerent license plates are there that involve 1, 2, or 3 letters
followed by 4 digits (with repetitions)?
Problem 31.3
a. In how many ways can 4 cards be drawn, with replacement, from a deck
of 52 cards?
b. In how many ways can 4 cards be drawn, without replacement, from a
deck of 52 cards?
Problem 31.4
In how many ways can 7 women and 3 men be arranged in a row if the three
men must always stand next to each other.
Problem 31.5
A menu in a Chinese restaurant allows you to order exactly two of eight
main dishes as part of the dinner special. How many diﬀerent combinations
of main dishes could you order?
Problem 31.6
Find the coeﬃcient of a5 b7 in the binomial expansion of (1 − 2b)12 .
Problem 31.7
Use the binomial theorem to prove that
n
n
3 = 2k C(n, k).
k=0
180.
180 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
32 Basic Probability Terms and Rules
Probability theory is one of the serious branch of mathematics with applica-
tions to many sciences, namely the theory of statistics. This section intro-
duces the most basic ideas of probabiltiy.
An experiment is any operation whose outcomes cannot be predicted with
certainty. The sample space S of an experiment is the set of all possible
outcomes for the experiment. For example, if you roll a die one time then
the experiment is the roll of the die. A sample space for this experiment is
S = {1, 2, 3, 4, 5, 6} where each digit represents a face of the die.
An event is any subset of a sample space. Thus, if S is the sample space
then the collection of all possible events is the power set P(S).
The Probability of an event E is the measure of occurrence of E. It is a
number between 0 and 1. If the event is impossible to occur then its proba-
bility is 0. If the occurrence is certain then the probability is 1. The closer
to 1 the probability is, the more likely the event is. The probability of oc-
currence of an event E (called its success) will be denoted by P (E). Thus,
0 ≤ P (E) ≤ 1. If an event has no outcomes, that is as a subset of S if E = ∅
then P (∅) = 0. On the other hand, if E = S then P (S) = 1.
Example 32.1
Which of the following numbers cannot be the probability of some event? (a)
4
0.71 (b)−0.5 (c) 150% (d) 3 .
Solution.
(a) Yes. (b) No. Since the number is negative. (c) No since the number is
greater than 1. (d) No.
Various probability concepts exist nowadays. The classical probability con-
cept applies only when all possible outcomes are equally likely, in which
case we use the formula
number of outcomes f avorable to event |E|
P (E) = = ,
total number of outcomes |S|
where |E| is the number of elements in E.
Example 32.2
What is the probability of drawing an ace from a well-shuﬂed deck of 52
playing cards?
181.
32 BASIC PROBABILITY TERMS AND RULES 181
Solution.
4 1
P (Ace) = 52 = 13
.
Example 32.3
What is the probability of rolling a 3 or a 4 with a fair die?
Solution.
2 1
P (3 or 4) = 6
= 3.
A major shortcoming of the classical probability concept is its limited applica-
bility, for there are many situations in which the various outcomes cannot
all regarded as equally likely. This would be the case, for instance, when
we wonder whether a person will get a raise or when we want to predict the
outcome of an election. A widely used probability concept is the estimated
probability which uses the relative frequency of an event and is given by the
formula:
f
P (E) = Relative f requency = ,
n
where f is the frequency of the event and n is the size of the sample space.
Example 32.4
Records show (over a period of time) that 468 of 600 jets from Dallas to
Phoenix arrived on time. Estimate the probability that any one jet from
Dallas to Phoenix will arrive on time.
Solution.
f 468 39
P (E) = n = 600
= 50
We deﬁne the probability of nonoccurrence of an event E (called its fail-
ure) by the formula
P (E c ) = 1 − P (E).
Note that
P (E) + P (E c ) = 1.
Example 32.5
The probability that a college student without a ﬂu shot will get the ﬂu is
0.45. What is the probability that a college student without the ﬂu shot will
not get the ﬂu?
182.
182 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Solution.
The probability is 1 − 0.45 = .55.
Next, we discuss some of the rules of probability. The union of two events
A and B is the event A ∪ B whose outcomes are either in A or in B. The
intersection of two events A and B is the event A ∩ B whose outcomes
are outcomes of both events A and B. Two events A and B are said to be
mutually exclusive if they have no outcomes in common. In this case
A ∩ B = ∅.
Example 32.6
If A and B are mutually exclusive then what is P (A ∩ B)?
Solution.
P (∅) = 0.
Theorem 32.1
For any events A and B the probability of A ∪ B is given by the addition
rule
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
If A and B are mutually exclusive then by Exercise 443 the above formula
reduces to
P (A ∪ B) = P (A) + P (B).
Proof.
By the Inclusion-Inclusion Principle we have
|A ∪ B| = |A| + |B| − |A ∩ B|.
Thus,
|A∪B|
P (A ∪ B) = |S|
|A|
= |S|
+ |B| − |A∩B|
|S| |S|
= P (A) + P (B) − P (A ∩ B).
Example 32.7
For any event E of a sample space S show that P (E) = x∈E P (x).
Solution.
This follows from the previous theorem
183.
32 BASIC PROBABILITY TERMS AND RULES 183
Example 32.8
M &M plain candies come in a variety of colors. According to the manufac-
turer, the color distribution is:
(a) Orange: 15% (b) Green: 10% (c) Red: 20% (d) Yellow: 20% (e)
Brown: 30% (f) Tan: 5%.
Suppose you have a large bag of plain candies and you reach in and take one
candy at random. Find
1. P(orange candy Or tan candy). Are these outcomes mutually exclusive?
2. P(not brown candy).
Solution.
1. P(orange candy Or tan candy) = .15 + .05 = .2 = 20%. The outcomes
are mutually exclusive.
2. P(not brown candy) = 1 − .3 = .7 = 70%
Example 32.9
If A is the event ”drawing an ace” from a deck of cards and B is the event
”drawing a spade”. Are A and B mutually exclusive? Find P (A ∪ B).
Solution.
The events are not mutually exclusive since there is an ace that is also a
spade.
4 13 1
P (A ∪ B) = P (A) + P (B) − P (A ∩ B) = + − = 31%
13 52 52
Now, given two events A and B belonging to the same sample space S.
The conditional probability P (A|B) denotes the probability that event A
will occur given that event B has occurred. It is given by the formula
P (A ∩ B)
P (A|B) = .
P (B)
Example 32.10
Consider the experiment of tossing two dice. What is the probability that
the sum of two dice equals six given that the ﬁrst die is a four?
184.
184 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Solution.
The possible outcomes of our experiment are
{(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)}.
Thus, the probability that the sum is six given that the ﬁrst die is four is 1 .
6
Assuming that the experiment consists of tossing the two dice then by letting
B be the event that the ﬁrst die is 4 and A be the even that the sum of the
two dice is 6 then
1
P (A ∩ B) 36 1
P (A|B) = = 6 =
P (B) 36
6
If P (A|B) = P (A), we say that the two events A and B are independent.
That is, the occurrence of A is independent whether or not B occurs. If two
events are not independent, we say that they are dependent.
Example 32.11
Show that A and B are independent if and only if
P (A ∩ B) = P (A) · P (B).
Solution.
Suppose that A and B are independent. Then P (A) = P (A|B) = P P (B) .(A∩B)
That is, P (A ∩ B) = P (A) · P (B). Conversely, if P (A ∩ B) = P (A) · P (B)
then P (A|B) = P P (B) = P (A).
(A∩B)
Example 32.12
You roll two fair dice: a green one and a red one.
a. Are the outcomes on the dice independent?
b. Find P(5 on green die and 3 on red die).
c. Find P(3 on green die and 5 on red die).
d. Find P((5 on green die and 3 on red die) or (3 on green die and 5 on red
die)).
Solution.
a. Yes.
b. P(5 on green die and 3 on red die) = 1 · 1 = 36 .
6 6
1
1
c. P(3 on green die and 5 on red die) = 36 .
d. P((5 on green die and 3 on red die) or (3 on green die and 5 on red die))
1 1 1
= 36 + 36 = 18 .
185.
32 BASIC PROBABILITY TERMS AND RULES 185
Example 32.13
Show that
P (B) · P (A|B)
P (B|A) = .
P (A)
Solution.
This follows from the fact that P (A ∩ B) = P (B ∩ A) and the formula of
P (A|B) given above.
Example 32.14
Prove Bayes’ Theorem
P (B|A)P (A)
P (A|B) = .
P (B|A)P (A) + P (B|Ac )P (Ac )
Solution.
Note ﬁrst that {Ac ∩ B, A ∩ B} form a partition of B. Thus,
P (B) = P (A ∩ B) + P (Ac ∩ B).
Now by the previous exercise we have
P (A)·P (B|A)
P (A|B) = P (B)
P (B|A)P (A)
= P (B|A)P (A)+P (B|Ac )P (Ac )
Example 32.15
Consider two urns. The ﬁrst contains two white and seven black balls and
the second contains ﬁve white and six black balls. We ﬂip a fair coin and
then draw a ball from the ﬁrst urn or the second urn depending on whether
the outcome was head or tail. What is the conditional probability that the
outcome of the toss was head given that a white ball was selected?
Solution.
Let W be the event that a white ball is drawn, and let H be the event that
the coin comes up heads. The desired probability P (H|W ) may be calculated
as follows:
P (H∩W )
P (H|W ) = P (W )
P (W |H)P (H)
= P (W )
P (W |H)P (H)
= P (W |H)P (H)+P (W |H c )P (H c )
2 1
22
= 21
9 2
5 1
+ 11 2
= 67
92
186.
186 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
It frequently occurs that in performing an experiment we are mainly inter-
ested in some functions of the outcome as opposed to the outcome itself. For
example, in tossing dice we are interested in the sum of the dice and are not
really concerned about the actual outcome. These real-valued functions de-
ﬁned on the sample space are known as random variables. If the range is
a ﬁnite subset of IN then the random variable is called discrete. Otherwise,
the random variable is said to be continuous. Discrete random variable are
usually the result of a count whereas a continuous random variable is usually
the result of a measurement.
A probability distribution is a correspondence that assigns probabilities
to the values of a random variable. The graph of a probability distribution
is called is a histogram.
Example 32.16
Let f denote the random variable that is deﬁned as the sum of two fair dice.
Find the probability distribution of f.
Solution.
1
P (f = 2) = P ({(1, 1)}) = 36 ,
2
P (f = 3) = P ({(1, 2), (2, 1)}) = 36 ,
3
P (f = 4) = P ({(1, 3), (2, 2), (3, 1)}) = 36 ,
4
P (f = 5) = P ({(1, 4), (2, 3), (3, 2), (4, 1)}) = 36 ,
5
P (f = 6) = P ({(1, 5, (5, 1), (2, 4), (4, 2), (3, 3)}) = 36 ,
6
P (f = 7) = P ({(1, 6), (6, 1), (2, 5), (5, 2), (4, 3), (3, 4)}) = 36
,
5
P (f = 8) = P ({(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)}) = 36 ,
4
P (f = 9) = P ({(3, 6), (6, 3), (4, 5), (5, 4)}) = 36 ,
3
P (f = 10) = P ({(4, 5), (5, 4), (5, 5)}) = 36 ,
2
P (f = 11) = P ({(5, 6), (6, 5)}) = 36 ,
1
P (f = 12) = P ({(6, 6)}) = 36 .
Example 32.17
Construct the histogram of the random variable of the previous exercise.
187.
32 BASIC PROBABILITY TERMS AND RULES 187
Solution.
For a discrete random variable f we deﬁne the expected value ( or mean)
of f by the formula
E(f ) = f (x)P (x)
x∈S
In other words, E(f ) is a weighted average of the possible values that f can
take on, each value being weighted by the probability that f assumes that
value.
Example 32.18
Find E(f ) where f is the outcome when we roll a fair die.
Solution.
1
Since P (1) = P (2) = · · · = P (6) = 6
we ﬁnd
1 1 1 7
E(f ) = 1( ) + 2( ) + · · · + 6( ) =
6 6 6 2
188.
188 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Another quantity of interest is the variance of a random variable f , de-
noted by V ar(f ), which is deﬁned by
V ar(f ) = E[(f − E(f ))2 ].
In other words, the variance measures the expected square of the deviation
of f from its expected value. The standard deviation of a random variable
f is the quantity is deﬁned to be the square root of the variance.
Example 32.19
Show that if f and g are random variables then E(f + cg) = E(f ) + cE(g)
where c is a constant.
Solution.
Indeed,
E(f + cg) = x∈S (f+ cg)(x)P (x)
= x∈S f (x)P (x) + c x∈S g(x)P (x)
= E(f ) + cE(g)
Theorem 32.2
V ar(f ) = E(f 2 ) − (E(f ))2 .
Proof.
Indeed, using the previous exercise we have
V ar(f ) = E(f 2 − 2E(f )f + (E(f ))2 )
= E(f 2 ) − 2E(f )E(f ) + (E(f ))2
= E(f 2 ) − (E(f ))2
Example 32.20
Calculate V ar(f ) when f represents the outcome when a fair die is rolled.
Solution.
First note that
91
E(f 2 ) = (f (1))2 P (1) + · · · + (f (6))2 P (6) = .
6
By the above theorem we have
91 7 35
V ar(f ) = E(f 2 ) − (E(f ))2 = − ( )2 =
6 2 12
189.
32 BASIC PROBABILITY TERMS AND RULES 189
Review Problems
Problem 32.1
What is the probability of drawing a red card from a well- shuﬄed deck of
52 playing cards?
Problem 32.2
If we roll a fair die, what are the probabilities of getting
a. a 1 or a 6;
b. an even number?
Problem 32.3
A department store’s records show that 782 of 920 women who entered the
store on a saturday afternoon made at least one purchase. Estimate the
probability that a woman who enters the store on a Saturday afternoon will
make at least one purchase.
Problem 32.4
Which of the following are mutually exclusive? Explain your answers.
a. A driver getting a ticket for speeding and a ticket for going through a red
light.
b. Being foreign-born and being President of the United States.
Problem 32.5
If A and B are the events that a consumer testing service will rate a given
stereo system very good or good, P (A) = 0.22, P (B) = 0.35. Find
a. P (Ac );
b. P (A ∪ B);
c. P (A ∩ B).
Problem 32.6
If the probabilities are 0.20, 0.15, and 0.03 that a student will get a failing
grade in Statistics, in English, or in both, what is the probability that the
student will get a failing grade in at least one of these subjects?
Problem 32.7
If the probability that a research project will be well planned is 0.60 and the
probability that it will be well planned and well executed is 0.54, what is the
probability that a well planned research project will be well executed?
190.
190 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
Problem 32.8
Given three events A, B, and C such that P(A)=0.50, P(B)=0.30, and P (A∩
B) = 0.15. Show that the events A and B are independent.
Problem 32.9
There are 16 equally likely outcomes by ﬂipping four coins. Let f repre-
sent the number of heads. Find the probability distribution and graph the
corresponding histogram.
191.
33 BINOMIAL RANDOM VARIABLES 191
33 Binomial Random Variables
In this section we discuss an important example of a discrete random variable.
Binomial experiments are problems that consist of a ﬁxed number of
trials n, with each trial having exactly two possible outcomes: Success and
failure. The probability of a success is denoted by p = P (S) and that of a
failure by q = P (F ). Moreover, p and q are related by the formula
p + q = 1.
Also, we assume that the trials are independent, that is what happens to
one trial does not aﬀect the probability of a success in any other trial. The
central question of a binomial experiment is to ﬁnd the probability of r suc-
cesses out of n trials. Now, anytime we make selections from a population
without replacement, we do not have independent trials. For example, se-
lecting a ball from a box that contain balls of two diﬀerent colors. If the
selection is without replacement then the trials are dependent.
Example 33.1
The registrar of a college noted that for many years the withdrawal rate from
an introductory chemistry course has been 35% each term. We wish to ﬁnd
the probability that 55 students out of 80 will register for the course.
a. What makes a trial?
b. What is a success? a failure?
c. What are the values of n, p, q, r?
Solution.
a. The decision of each student to withdraw or complete the course can be
thought as a trial. Thus, there are a total of 80 trials.
b. S = completing the course, F = withdrawing from course.
c. n = 80, p = .65, q = .35, r = 55.
Example 33.2
Harper’s Index states that 10% of all adult residents in Washington D.C.,
are lawyers. For a random sample of 15 adult Washington, D.C., residents,
we want to ﬁnd the probability that 3 are lawyers.
a. What makes a trial?
192.
192 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
b. What is a success? a failure?
c. What are the values of n, p, q, r?
Solution.
a. A trial is whether an adult resident of Washington, D.C. is a lawyer or
not.
b. S = being a lawyer, F = not being a lawyer.
c. n = 15, p = .1, q = .9, r = 3.
As mentioned earlier, the central problem of a binomial experiment is to
ﬁnd the probability of r successes out of n independent trials. We next see
how to ﬁnd these probabilities.
Recall from Section 6.1 the formula for ﬁnding the number of combinations
of n distinct objects taken r at a time
n!
C(n, r) = .
r!(n − r)!
We call the number C(n, r) the binomial coeﬃcient. One commonly used
procedure for ﬁnding these coeﬃcients is by means of Pascal’s triangle.
Now, the probability of r successes out of n independent trials is given by
the binomial distribution formula
P (r) = C(n, r)pr q n−r
where p = P (S) and q = P (F ) = 1 − p. The validity of the above equation
may be veriﬁed by ﬁrst noting that the probability of any particular sequence
of the n outcomes with r successes and n − r failures is, by the independence
of trials, pr (1−p)n−r . Since C(n, r) counts the number of outcomes that have
r successes and n − r failures then the equation above follows.
Example 33.3
Find the probability that in tossing a fair coin three times there will appear
(a) 3 heads, (b)2 heads and 1 tail, (c) 2 tails and 1 head, and (d) 3 tails.
Solution.
a. C(3, 3)(.5)3 (.5)3−3 = 1 .
8
3
b. P (2) = C(3, 2)(.5)2 (.5) = 8 .
2 3
c. P (1) = C(3, 1)(.5)(.5) = 8 .
1
d. P (0) = C(3, 0)(.5)3 = 8
193.
33 BINOMIAL RANDOM VARIABLES 193
Example 33.4
The probability that an entering college student will graduate is 0.4. Deter-
mine the probability that out of 5 students (a) none, (b) 1, (c) at least 1, (d)
all will graduate.
Solution.
(a) C(5, 0)(.6)5 .
(b) C(5, 1)(.4)(.6)4 .
(c) 1 − C(5, 0)(.6)5 .
(d) C(5, 5)(.4)5 .
Example 33.5
Find the probability of guessing correctly at least 6 of the 10 answers on a
true-false examination.
Solution.
P (6) + P (7) + P (8) + P (9) + P (10).
We next derive formulas for ﬁnding the expected value and standard de-
viation for the binomial random variable.
Theorem 33.1
a. The mean of a binomial random variable is given by µ = np.
b. The variance of a binomial random variable is given by σ 2 = npq.
Proof.
a. Using the deﬁnition of µ we have
n
µ = i=0 iP (i)
n i n−i
= i=1 iC(n, i)p q
n n!
= np i=1 (i−1)!(n−i) pi−1 q n−i−1
n−1
= np i=0 C(n − 1, i)pi q n−1−i
= np(p + q)n−1 = np.
194.
194 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
b. Note ﬁrst that i2 = i(i − 1) + i. Then
n
E(X 2 ) = 2
i=0 i P (i)
n i n−i
= i=0 i(i − 1)C(n, i)p q +µ
n n!
= i=2 (n−i)!(i−2)! pi q n−i + µ
= n(n − 1)p2 n (n−i)!(i−2)! pi q n−i + µ
i=2
(n−2)!
= n(n − 1)p2 n−2 C(n − 2, j)pj q n−2−j + µ
j=0
= n(n − 1)p2 (p + q)n−2 + µ
= n(n − 1)p2 + µ
It follows that
σ2 = E(X 2 ) − µ2
= n(n − 1)p2 + np − n2 p2
= npq
195.
33 BINOMIAL RANDOM VARIABLES 195
Review Problems
Problem 33.1
At Community Hospital, the nursing staﬀ is large enough so that 80% of the
time a nurse can respond to a room call within 3 minutes. Last night there
were 73 room calls. We wish to ﬁnd the probability nurses responded to 62
of them within 3 minutes.
a. What makes a trial?
b. What is a success? a failure?
c. What are the values of n, p, q, r?
Problem 33.2
Find the probability that in a family of 4 children there will be (a) at least 1
boy and (b) at least 1 boy and 1 girl. Assume that the probability of a male
birth is 1 .
2
Problem 33.3
An insurance salesperson sells policies to 5 men, all of identical age and in
good health. According to the actuarial tables, the probability that a man
2
of this particular age will be alive 30 years is 3 . Find the probability that in
30 years (a) all 5 men, (b) at least 3 men, (c) only 2 men, (d) none will be
alive.
196.
196 FUNDAMENTALS OF COUNTING AND PROBABILITY THEORY
197.
Elements of Graph Theory
In this chapter we present the basic concepts related to graphs and trees such
as the degree of a vertex, connectedness, Euler and Hamiltonians circuits,
isomorphisms of graphs, rooted and spanning trees.
34 Graphs, Paths, and Circuits
An undirected graph G consists of a set VG of vertices and a set EG of
edges such that each edge e ∈ EG is associated with an unordered pair of
vertices, called its endpoints.
A directed graph or digraph G consists of a set VG of vertices and a set
EG of edges such that each edge e ∈ EG is associated with an ordered pair
of vertices.
We denote a graph by G = (VG , EG ).
Two vertices are said to be adjacent if there is an edge connecting the two
vertices. Two edges associated to the same vertices are called parallel. An
edge incident to a single vertex is called a loop. A vertex that is not incident
on any edge is called an isolated vertex. A graph with neither loops nor
parallel edges is called simple graph.
Example 34.1
Consider the following graph G
197
198.
198 ELEMENTS OF GRAPH THEORY
a. Find EG and VG .
b. List the isolated vertices.
c. List the loops.
d. List the parallel edges.
e. List the vertices adjacent to v3 ..
f. Find all edges incident on v4 .
Solution.
a. EG = {e1 , e2 , e3 , e4 , e5 , e6 } and VG = {v1 , v2 , v3 , v4 , v5 , v6 , v7 }.
b. There is only one isolated vertex, v5 .
c. There is only one loop, e5 .
d. {e2 , e3 }.
e. {v2 , v4 }.
f. {e1 , e4 , e5 }.
Example 34.2
Which one of the following graphs is simple.
Solution.
a. G is not simple since it has a loop and parallel edges.
b. G is simple.
A complete graph on n vertices, denoted by Kn , is the simple graph that
contains exactly one edge between each pair of distinct vertices.
Example 34.3
Draw K2 , K3 , K4 , and K5 .
199.
34 GRAPHS, PATHS, AND CIRCUITS 199
Solution.
A graph in which the vertices can be partitioned into two disjoint sets V1
and V2 with every edge incident on one vertex in V1 and one vertex of V2 is
called bipartite graph.
Example 34.4
a. Show that the graph G is bipartite.
b. Show that K3 is not bipartite.
Solution.
a. Clear from the deﬁnition and the graph.
b. Any two sets of vertices of K3 will have opposite parity. Thus, according
to the deﬁnition of bipartite graph, K3 is not bipartite.
A complete bipartite graph Km,n , is the graph that has its vertex set
partitioned into two disjoint subsets of m and n vertices, respectively. More-
200.
200 ELEMENTS OF GRAPH THEORY
over, there is an edge between two vertices if and only if one vertex is in the
ﬁrst set and the other vertex is in the second set.
Example 34.5
Draw K2,3 , K3,3 .
Solution.
The degree of a vertex v in an undirected graph, in symbol deg(v), is the
number of edges incident on it. By deﬁnition, a loop at a vertex contributes
twice to the degree of that vertex. The total degree of G is the sum of the
degrees of all the vertices of G.
Example 34.6
What are the degrees of the vertices in the following graph
Solution.
deg(v1 ) = 0, deg(v2 ) = 2, deg(v3 ) = 4.
201.
34 GRAPHS, PATHS, AND CIRCUITS 201
Theorem 34.1
For any graph G = (VG , EG ) we have
2|EG | = deg(v).
v∈V (G)
Proof.
Suppose that VG = {v1 , v2 , · · · , vn } and |EG | = m. Let e ∈ EG . If e is a loop
then it contributes 2 to the total degree of G. If e is not a loop then let vi and
vj denote the endpoints of e. Then e contributes 1 to deg(vi ) and contributes
1 to the deg(vj ). Therefore, e contributes 2 to the total degree of G. Since
e was arbitrarily, this shows that each edge of G contributes 2 to the total
degree of G. Thus,
2|EG | = deg(v)
v∈V (G)
The following is easily deduced from the previous theorem.
Theorem 34.2
In any graph there are an even number of vertices of odd degree.
Proof.
Let G = (VG , EG ) be a graph. By the previous theorem, the sum of all the
degrees of the vertices is T = 2|EG |, an even number. Let E be the sum of
the numbers deg(v), each which is even and O the sum of numbers deg(v)
each which is odd. Then T = E + O. That is, O = T − E. Since both T
and E are even then T is also even. This implies that there must be an even
number of the odd degrees. Hence, there must be an even number of vertices
with odd degree.
Example 34.7
Find a formula for the number of edges in Kn .
Solution.
Since G is complete, each vertex is adjacent to the remaining vertices. Thus,
the degree of each of the n vertices is n − 1, and we have the sum of the de-
grees of all of the vertices being n(n−1). By Theorem 34.1, n(n−1) = 2|EG |.
This completes a proof of the theorem
202.
202 ELEMENTS OF GRAPH THEORY
In an undirected graph G a sequence P of the form v0 e1 v1 e2 · · · vn−1 en vn
is called a path of length n or a path connecting v0 to vn . If P is a path
such that v0 = vn then it is called a circuit or a cycle. A path or circuit is
simple if it does not contain the same edge more than once. A graph that
does not contain any circuit is called acyclic.
Example 34.8
In the graph below, determine whether the following sequences are paths,
simple paths, circuits, or simple circuits.
a. v0 e1 v1 e10 v5 e9 v2 e2 v1 .
b. v3 e5 v4 e8 v5 e10 v1 e3 v2 .
c. v1 e2 v2 e3 v1 .
d. v5 e9 v2 e4 v3 e5 v4 e6 v4 e8 v5 .
Solution.
a. a path (no repeated edge), not a simple path (repeated vertex v1 ), not a
circuit
b. a simple path
c. a simple circuit
d. a circuit, not a simple circuit (vertex v4 is repeated)
An undirected graph is called connected if there is a path between every
pair of distinct vertices of the graph. A graph that is not connected is said
to be disconnected.
203.
34 GRAPHS, PATHS, AND CIRCUITS 203
Example 34.9
Determine which graph is connected and which one is disconnected.
Solution.
a. Connected.
b. Disconnected since there is no path connecting the vertices v1 and v4 .
A simple path that contains all edges of a graph G is called an Euler path.
If this path is also a circuit, it is called an Euler circuit.
Theorem 34.3
If a graph G has an Euler circuit then every vertex of the graph has even
degree.
Proof.
Let G be a graph with an Euler circuit. Start at some vertex on the circuit
and follow the circuit from vertex to vertex, erasing each edge as you go
along it. When you go through a vertex you erase one edge going in and one
edge going out, or else you erase a loop. Either way, the erasure reduces the
degree of the vertex by 2. Eventually every edge gets erased and all the ver-
tices have degree 0. So all vertices must have had even degree to begin with.
It follows from the above theorem that if a graph has a vertex with odd
degree then the graph can not have an Euler circuit.
The following provide a converse to the above theorem.
204.
204 ELEMENTS OF GRAPH THEORY
Theorem 34.4 (Euler Theorem)
If all the vertices of a connected graph have even degree, then the graph has
an Euler circuit.
Example 34.10
Show that the following graph has no Euler circuit.
Solution.
Vertices v1 and v3 both have degree 3, which is odd. Hence, by the remark
following the previous theorem, this graph does not have an Euler circuit.
A path is called a Hamiltonian path if it visits every vertex of the graph
exactly once. A circuit that visits every vertex exactly once except for the
last vertex which duplicates the ﬁrst one is called a Hamiltonian circuit.
Example 34.11
Find a Hamiltonian circuit in the graph
Solution.
vwxyzv
205.
34 GRAPHS, PATHS, AND CIRCUITS 205
Example 34.12
Show that the following graph has a Hamiltonian path but no Hamiltonian
circuit.
Solution.
vwxyz is a Hamiltonian path. There is no Hamiltonian circuit since no cycle
goes through v.
206.
206 ELEMENTS OF GRAPH THEORY
Review Problems
Problem 34.1
The union of two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) is the graph
G1 ∪ G2 = (V1 ∪ V2 , E1 ∪ E2 ). The intersection of two graphs G1 = (V1 , E1 )
and G2 = (V2 , E2 ) is the graph G1 ∩ G2 = (V1 ∩ V2 , E1 ∩ E2 ).
Find the union and the intersection of the graphs
Problem 34.2
Graphs can be represented using matrices. The adjacency matrix of a graph
G with n vertices is an n×n matrix AG such that each entry aij is the number
of edges connecting vi and vj . Thus, aij = 0 if there is no edge from vi to vj .
a. Draw a graph with the adjacency matrix
0 1 1 0
1 0 0 1
1 0 0 1
0 1 1 0
b. Use an adjacency matrix to represent the graph
207.
34 GRAPHS, PATHS, AND CIRCUITS 207
Problem 34.3
A graph H = (VH , EH ) is a subgraph of G = (VG , EG ) if and only if VH ⊆ VG
and EH ⊆ EG .
Find all nonempty subgraphs of the graph
When (u, v) is an edge in a directed graph G then u is called the ini-
tial vertex and v is called the terminal vertex. In a directed graph,
the in-degree of a vertex v, denoted by deg − (v), is the number of edges
with v as their terminal vertex. Similarly, the out-degree of v, denoted
by deg + (v), is the number of edges with v as an initial vertex. Note that
deg(v) = deg + (v) + deg − (v).
208.
208 ELEMENTS OF GRAPH THEORY
Problem 34.4
Find the in-degree and out-degree of each of the vertex in the graph G with
directed edges.
Problem 34.5
Show that for a digraph G = (VG , EG ) we have
|EG | = deg − (v) = deg + (v).
v∈V (G) v∈V (G)
Another useful matrix representation of a graph is known as the incidence
matrix. It is constructed as follows. We label the rows with the vertices
and the columns with the edges. The entry for row v and column e is 1 if e
is incident on v and 0 otherwise. If e is a loop at v we assign the value 2. It
is easy to see that the sum of entries of each column is 2 and that the sum
of entries of a row gives the degree of the vertex corresponding to that row.
Problem 34.6
Find the incidence matrix corresponding to the graph
209.
34 GRAPHS, PATHS, AND CIRCUITS 209
Problem 34.7
If each vertex of an undirected graph has degree k then the graph is called a
regular graph of degree k.
How many edges are there in a graph with 10 vertices each of degree 6?
Problem 34.8
Two simple graphs G1 and G2 are isomorphic, in symbol, G1 G2 , if there
is one-to-one onto function, f : V (G1 ) → V (G2 ) and AG1 = AG2 . Show that
the following graphs are isomorphic.
Warning: The number of vertices, the number of edges, and the degrees of
the vertices are all invariants under isomorphism. If any of these quantities
210.
210 ELEMENTS OF GRAPH THEORY
diﬀer in two graphs, these graphs cannot be isomorphic. However, when these
invariants are the same, it does not necessarily mean that the two graphs are
isomorphic.
The isomorphism between two graphs G1 = (VG1 , EG1 ) and G2 = (VG2 , EG2 )
with parallel edges or loops requires two bijections f : VG1 → VG2 and g :
EG1 → EG2 such that if e ∈ EG1 is an edge with endpoints (u, v) then
g(e) ∈ EG2 is an edge with endpoints (f (u), f (v)).
Problem 34.9
Show that the following graphs are not isomorphic.
Problem 34.10
Show that the following graph has no Hamiltonian path.
211.
35 TREES 211
35 Trees
An undirected graph is called a tree if each pair of distinct vertices has ex-
actly one path. Thus, a tree has no parallel edges and no loops.
We next show a result that is needed for the proof of our ﬁrst main theorem
of trees.
Theorem 35.1
Any tree with more than one vertex has at least one vertex of degree 1.
Proof.
Let v0 and vn be two distinct vertices. Then there is a path connecting v0 to
vn . By the deﬁnition of a tree, there is only one edge incident on either v0 or
vn . Thus deg(v0 ) = deg(v1 ) = 1. .
The following is the ﬁrst of the two main theorems about trees:
Theorem 35.2
A tree with n vertices has exactly n − 1 edges.
Proof.
The proof is by induction on n ≥ 1. Let P (n) be the property: Any tree with
n vertices has n − 1 edges.
Basis of induction: P (1) is valid since a tree with one vertex has zero edges.
Induction hypothesis: Suppose that P (n) holds up to n ≥ 1.
Induction Step: We must show that any tree with n + 1 vertices has n edges.
Indeed, let T be any tree with n + 1 vertices. Since n + 1 ≥ 2 then by the
previous theorem, T has a vertex v of degree 1. Let T0 be the graph obtained
by removing v and the edge attached to v. Then T0 is a tree with n vertices.
By the induction hypothesis, T0 has n − 1 edges and so T has n edges
Example 35.1
Which of the following graphs are trees?
212.
212 ELEMENTS OF GRAPH THEORY
Solution.
The ﬁrst graph satisﬁes the deﬁnition of a tree. The second and third graphs
do not satisfy the conclusion of Theorem 35.2 and therefore they are not
trees.
The second major theorem about trees is the following theorem whose proof
is omitted.
Theorem 35.3
Any connected graph with n vertices and n − 1 edges is a tree.
A rooted tree is a tree in which a particular vertex is designated as the
root. The level of a vetex v is the length of the simple path from the root
to v. The height of a rooted tree is the maximum level number that occurs.
Example 35.2
Find the level of each vertex and the height of the following rooted tree.
213.
35 TREES 213
Solution.
v1 is the root of the given tree.
vertex level
v2 1
v3 1
v4 2
v5 2
v6 2
v7 2
The height of the tree is 2.
Let T be a rooted tree with root v0 . Suppose (v0 , v1 , · · · , vn ) is a simple
path in T and x, y, z are three vertices. Then
(a) vn−1 is the parent of vn .
(b) v0 , v1 , · · · , vn−1 are the ancestors of vn .
(c) vn is the child of vn−1 .
(d) If x is an ancestor of y then y is a descendant of x.
(e) If x and y are children of z then x and y are siblings.
(f) If x has no children, then x is a leaf.
(g) The subtree of T rooted at x is the graph with vertex V and edge set
E, where V is x together with the descendants of x and
E = {e|e is an edge on a simple path f rom x to some vertex in V }.
Example 35.3
Consider the rooted tree
214.
214 ELEMENTS OF GRAPH THEORY
a. Find the parent of v6 .
b. Find the ancestors of v13 .
c. Find the children of v3 .
d. Find the descendants of v11 .
e. Find an example of a siblings.
f. Find the leaves.
g. Construct the subtree rooted at v7 .
Solution.
a. v2 .
b. v1 , v3 , v7 .
c. v7 , v8 , v9 .
d. None.
e. {v2 , v3 , v4 , v5 }.
f. {v4 , v5 , v6 , v9 , v9 , v10 , v11 , v12 , v13 }.
g.
A binary tree is a rooted tree such that each vertex has at most two chil-
dren. Moreover, each child is designated as either a left child or a right
child.
Example 35.4
a. Show that the following tree is a binary tree.
215.
35 TREES 215
b. Find the left child and the right child of vertex v5 .
c. A full binary tree is a binary tree in which each vertex has either two
children or zero children. Construct an example of a full binary tree.
Solution.
a. Follows from the deﬁnition of a binary tree.
b. The left child is v6 and the right child is v7 .
c.
Example 35.5
A forest is a simple graph with no circuits. Which of the following graphs
is a forest?
216.
216 ELEMENTS OF GRAPH THEORY
Solution.
The ﬁrst graph is a forest whereas the second is not.
Example 35.6
a. Let T be a subgraph of a graph G such that T is a tree containing all of
the vertices of G. Such a tree is called a spanning tree. Find a spanning
tree of the following graph.
b. The following algorithm ﬁnds a spanning tree. In this algorithm S denotes
a sequence. Let G be a connected graph with vertices ordered
v 1 , v 2 , , · · · , vn
1. Let T be the tree with root v1 and no edges.
2. Add to T all edges (v1 , x) and vertices on which they are incident, provided
217.
35 TREES 217
that (v1 , x) deos not produce a circuit. If no edges can be added, stop (T is
a spanning tree)
3. Replace S by the children in T of S ordered consistently with the original
ordering. Go to step 2.
Use the above algorithm to ﬁnd the spanning tree of part a.
Solution.
a.
218.
218 ELEMENTS OF GRAPH THEORY
Review Problems
Problem 35.1
Find the level of each vertex and the height of the following rooted tree.
Problem 35.2
Consider the rooted tree
a. Find the parent of v6 .
b. Find the ancestors of v10 .
219.
35 TREES 219
c. Find the children of v4 .
d. Find the descendants of v1 .
e. Find all the siblings.
f. Find the leaves.
g. Construct the subtree rooted at v1 .
Problem 35.3
The binary tree below gives an algorithm for choosing a restaurant. Each
internal vertex asks a question. If we begin at the root, answer each ques-
tion, and follow the appropriate edge, we will eventually arrive at a terminal
vertex that chooses a restaurant. Such a tree is called a decision tree.
Construct a decision tree that sorts three given numbers a1 , a2 , a3 in as-
cending order.
Problem 35.4
A binary search tree is a binary tree T in which data are associated with
the vertices. The data are arranged so that, for each vertex v in T, each data
item in the left subtree of v is less than the data item in v and each data item
in the right subtree of v is greater than the data item in v. Using numerical
order, form a binary search tree for a number in the set {1, 2, · · · , 15}.
Problem 35.5
Procedures for systematically visiting every vertex of a tree are called tra-
versal algorithms. In the preorder traversal, the root r is listed ﬁrst
220.
220 ELEMENTS OF GRAPH THEORY
and then the subtrees T1 , T2 , · · · , Tn are listed, from left to right, in order
of their roots. The preorder traversal begins by visiting r. It continues by
traversing T1 in preorder, thenT2 in preorder, and so on, until Tn is traversed
in preorder. In which order does a preorder traversal visit the vertices in the
following rooted tree?
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